After understanding this answer, you will realize there are a lot of sutta, which explaining 3 Taṇhā with 4 Upādāna. You can use this explanation in everywhere of Tipiṭaka, i.e. brahmajālasutta because the sutta I linked below is just the example.
What is the difference between Tanha and Upadana?
Taṇhā is beginning-lobha, Upādāna is often-arising-lobha after that Taṇhā. Taṇhā is like a slightly touching-hand; Upādāna is like that hand which becomes a tight grip hand.
Taṇhā condition Upādāna, Upādāna condition Bhava, so everything, between Taṇhā until Bhava, is Upādāna and Bhava, such as in Sutta. Dī. Ma. Mahānidānasutta, which I will quote below.
Why Buddha did not say Tanha Paccaya Bhava (instead of Upadana Paccaya Bhava)?
Because just a beginning attachment (Taṇhā) is not enough to build Bhava, actually, often and various attachments are required to build Bhava. Such as in Sutta. Dī. Ma. Mahānidānasutta, Dependent on Craving Section (which is explained as Sutta. Ma. Mul. Mahādukkhakkhandhasutta):
(There are many explanations in those 2 links which I made above, please see inside them for the information between Taṇhā until Bhava.)
"Now, craving [Taṇhā] is dependent on feeling, seeking [Upādāna] is dependent on craving, acquisition [Upādāna] is dependent on seeking, ascertainment [Upādāna] is dependent on acquisition, desire and passion [Upādāna] is dependent on ascertainment, attachment [Upādāna] is dependent on desire and passion, possessiveness [Upādāna] is dependent on attachment, stinginess [dosa-typed mind factor] is dependent on possessiveness, defensiveness [Sīlabbata-Upādāna & kamma-Bhava] is dependent on stinginess, (the example of defensiveness:) and because of defensiveness, dependent on defensiveness, various evil, unskillful phenomena come into play: the taking up of sticks and knives; conflicts, quarrels, and disputes; accusations, divisive speech, and lies.
Note: This is the reason why sotāpanna cannot birth in apāya anymore because sotāpanna doesn't have Sīlabbata-Upādāna which causes of breaking precept like in the end effect of above quote.
If you can not understand Upādāna from above quote and it's the explanation in Mahānidānasutta, Dependent on Craving Section, please see Mahādukkhakkhandhasutta, which fully explaining about Upādāna until karma-Bhava.
And there are three kinds of Tanha in Sutta (Kama, Bhava, Vibhava): please explain how these three link to Upadana?
In Mahādukkhakkhandhasutta kāma-Taṇhā, which caving "five strands of sensuality", conditions Bhava/Vibhava-Taṇhā, then these 3 Taṇhās condition Upādāna-until-Jāti:
Sights known by the eye that are likable, desirable, agreeable, pleasant, sensual, and arousing.
Cakkhuviññeyyā rūpā iṭṭhā kantā manāpā piyarūpā (samudaya-saccaniddesa in saccapabba of mahāsatipaṭṭhānasutta) kāmūpasaṃhitā rajanīyā
After above quote, whole left Mahādukkhakkhandhasutta's content are Bhava/Vibhava-Taṇhā which condition Upādāna-until-Jāti.
Bhava-taṅhā&Vibhava-Taṇhā are connect to Diṭṭhi/Sīlabbata/Attavāda-Upādāna directly in Sutta. Ma. Mū. Cūlasīhanādasutta:
"Bhikkhus, there are these two views: the view of being (bhava-Diṭṭhi) and the view of non-being (Vibhava-Diṭṭhi). Any recluses or brahmans who rely on the view of being, adopt the view of being, accept the view of being, are opposed to the view of non-being. Any recluses or brahmans who rely on the view of non-being, adopt the view of non-being, accept the view of non-being, are opposed to the view of being.5
"Any recluses or brahmans who do not understand as they actually are the origin, the disappearance, the gratification, the danger (the same as in Mahādukkhakkhandhasutta because they are connect to each other) and the escape6 in the case of these two views are affected by lust, affected by hate, affected by delusion, affected by craving, affected by clinging, without vision, given to favoring and opposing, and they delight in and enjoy proliferation. They are not freed from birth, aging and death, from sorrow, lamentation, pain, grief and despair; they are not freed from suffering, I say.
- "Bhikkhus, there are these four kinds of clinging (Upādāna). What four? Clinging to sensual pleasures, clinging to views, clinging to rules and observances, and clinging to a doctrine of self.
Actually, 3 Taṇhā come together: you want to feel good (kāma), so you want to born as a feel-good-able person (bhava); And you don't want to born as a feel-bad-able person (Vibhava), because you don't want to feel bad.
Then kāma-Taṇhā condition kāma-Upādāna; Bhava/Vibhava-Taṇhā condition uccheda/sassata-Diṭṭhi-Upādāna & Sīlabbata-Upādāna, which depending on Attavāda-Upādāna. Please see the whole Sutta. Ma. Mū. Cūlasīhanādasutta, it is very clear inside that sutta.
Atthakathā/abhidhamma-teacher just connected sutta together. The reader will misunderstand of Tipiṭaka and Atthakathā, when they still think "these sutta are conflicting those sutta". In fact, they are studying Tipiṭaka by the inefficient study system, reading, so they feel like Tipiṭaka is conflicting each other.
After this I will let you see "you can use this explanation in everywhere of Tipiṭaka".
Sutta. Saṃ. Ma. Dhammacakkappavattanasutta:
Idaṃ kho pana, bhikkhave, dukkhaṃ ariya·saccaṃ: Jāti·pi dukkhā, jarā·pi dukkhā (byādhi·pi dukkho) maraṇam·pi dukkhaṃ, a·p·piyehi sampayogo dukkho, piyehi vippayogo dukkho, yampicchaṃ na labhati tam·pi dukkhaṃ; saṃkhittena pañc·Upādāna·k·khandhā dukkhā.
Furthermore, bhikkhus, this is the dukkha ariya·sacca(2): Jāti is dukkha(6), jarā is dukkha(6) (sickness is dukkha) maraṇa is dukkha(6), association with what is disliked is dukkha, dissociation from what is liked is dukkha, not to get what one wants is dukkha; in short, the five Upādāna(4)'k'khandhas are dukkha(3).
Idaṃ kho pana, bhikkhave, dukkha·samudayaṃ ariya·saccaṃ: Y·āyaṃ Taṇhā ponobbhavikā nandi·rāga·sahagatā tatra·tatr·ābhinandinī, seyyathidaṃ: kāma·Taṇhā, Bhava·Taṇhā, Vibhava·Taṇhā.
Furthermore, bhikkhus, this is the dukkha·samudaya ariya·sacca(1): this Taṇhā leading to rebirth(7), (because Taṇhā) connected with desire and enjoyment (8) (this is Upādāna; see Upādāna description in Mahānidānasutta, which I quoted above), (by Taṇhā) finding delight here or there(=idiom:indefinite=Upādāna)(5), that is to say: kāma-Taṇhā, Bhava-Taṇhā and Vibhava-Taṇhā.
Above, buddha taught dukkha·samudayaṃ ariya·saccaṃ Taṇhā condition dukkha·ariya·saccaṃ(2) pañc·Upādāna·k·khandhā dukkhā by Upādāna(4) [often attaching/Taṇhā finding delight here or there(5)] in khandhā dukkhā(3); therefore Jāti·pi dukkhā/jarā·pi dukkhā/maraṇam·pi dukkhaṃ(6) [new upatti-Bhava], which are the last effects in paṭiccasamuppāda, are included in dukkha·ariya·saccaṃ(2) and they are explained as new upatti-Bhava in this statement "Taṇhā leading to rebirth(new upatti-Bhava)"(7), which are the same Taṇhā's descriptions as it's follow statements "Taṇhā connected with desire and enjoyment(8)" and "Taṇhā finding delight here or there(5)".
Summary: (6)=(3) which depend on (4); and (4) is explained by (5), so when say (5) condition (7), it is the same as say (4) condition (7). Therefore buddha taught Upādāna-khandha, instead of Taṇhā-khandha.