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Theorem cbvabdavw 36199
Description: Change bound variable in class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvabdavw.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvabdavw (𝜑 → {𝑥𝜓} = {𝑦𝜒})
Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvabdavw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbvabdavw.1 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21cbvsbdavw 36197 . . 3 (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑦]𝜒))
3 df-clab 2711 . . 3 (𝑡 ∈ {𝑥𝜓} ↔ [𝑡 / 𝑥]𝜓)
4 df-clab 2711 . . 3 (𝑡 ∈ {𝑦𝜒} ↔ [𝑡 / 𝑦]𝜒)
52, 3, 43bitr4g 314 . 2 (𝜑 → (𝑡 ∈ {𝑥𝜓} ↔ 𝑡 ∈ {𝑦𝜒}))
65eqrdv 2731 1 (𝜑 → {𝑥𝜓} = {𝑦𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  [wsb 2060  wcel 2104  {cab 2710
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-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
This theorem is referenced by:  cbvsbcdavw  36200  cbvsbcdavw2  36201  cbvrabdavw  36204  cbviotadavw  36212  cbvixpdavw  36221  cbvrabdavw2  36228  cbvixpdavw2  36237
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