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Theorem cbvrabdavw2 36228
Description: Change bound variable and domain in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvrabdavw2.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
cbvrabdavw2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvrabdavw2 (𝜑 → {𝑥𝐴𝜓} = {𝑦𝐵𝜒})
Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrabdavw2
StepHypRef Expression
1 eleq1w 2820 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21adantl 481 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
3 cbvrabdavw2.2 . . . . . 6 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
43eleq2d 2823 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝑦𝐴𝑦𝐵))
52, 4bitrd 279 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
6 cbvrabdavw2.1 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
75, 6anbi12d 631 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
87cbvabdavw 36199 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑦 ∣ (𝑦𝐵𝜒)})
9 df-rab 3433 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
10 df-rab 3433 . 2 {𝑦𝐵𝜒} = {𝑦 ∣ (𝑦𝐵𝜒)}
118, 9, 103eqtr4g 2798 1 (𝜑 → {𝑥𝐴𝜓} = {𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  {cab 2710  {crab 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433
This theorem is referenced by: (None)
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