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Theorem cbvmptdavw 36246
Description: Change bound variable in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvmptdavw.1 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbvmptdavw (𝜑 → (𝑥𝐴𝐵) = (𝑦𝐴𝐶))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvmptdavw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2823 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21adantl 481 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
3 cbvmptdavw.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
43eqeq2d 2747 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡 = 𝐵𝑡 = 𝐶))
52, 4anbi12d 632 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝑡 = 𝐵) ↔ (𝑦𝐴𝑡 = 𝐶)))
65cbvopab1davw 36243 . 2 (𝜑 → {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐴𝑡 = 𝐵)} = {⟨𝑦, 𝑡⟩ ∣ (𝑦𝐴𝑡 = 𝐶)})
7 df-mpt 5224 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐴𝑡 = 𝐵)}
8 df-mpt 5224 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑡⟩ ∣ (𝑦𝐴𝑡 = 𝐶)}
96, 7, 83eqtr4g 2801 1 (𝜑 → (𝑥𝐴𝐵) = (𝑦𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {copab 5203  cmpt 5223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5204  df-mpt 5224
This theorem is referenced by:  cbvproddavw  36259  cbvitgdavw  36260  cbvproddavw2  36275  cbvitgdavw2  36276
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