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Theorem cbvcsbdavw 36202
Description: Change bound variable of a proper substitution into a class. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvcsbdavw.1 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbvcsbdavw (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶)
Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvcsbdavw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbvcsbdavw.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
21eleq2d 2823 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡𝐵𝑡𝐶))
32cbvsbcdavw 36200 . . 3 (𝜑 → ([𝐴 / 𝑥]𝑡𝐵[𝐴 / 𝑦]𝑡𝐶))
43abbidv 2804 . 2 (𝜑 → {𝑡[𝐴 / 𝑥]𝑡𝐵} = {𝑡[𝐴 / 𝑦]𝑡𝐶})
5 df-csb 3909 . 2 𝐴 / 𝑥𝐵 = {𝑡[𝐴 / 𝑥]𝑡𝐵}
6 df-csb 3909 . 2 𝐴 / 𝑦𝐶 = {𝑡[𝐴 / 𝑦]𝑡𝐶}
74, 5, 63eqtr4g 2798 1 (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1535  wcel 2104  {cab 2710  [wsbc 3791  csb 3908
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-sbc 3792  df-csb 3909
This theorem is referenced by:  cbvsumdavw  36222  cbvproddavw  36223  cbvsumdavw2  36238  cbvproddavw2  36239
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