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Theorem cbvcsbvw2 36174
Description: Change bound variable of a proper substitution into a class using implicit substitution. General version of cbvcsbv 3920. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
cbvcsbvw2.1 𝐴 = 𝐵
cbvcsbvw2.2 (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
cbvcsbvw2 𝐴 / 𝑥𝐶 = 𝐵 / 𝑦𝐷
Distinct variable groups:   𝑥,𝑦   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvcsbvw2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbvcsbvw2.1 . . . 4 𝐴 = 𝐵
2 cbvcsbvw2.2 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
32eleq2d 2823 . . . 4 (𝑥 = 𝑦 → (𝑡𝐶𝑡𝐷))
41, 3cbvsbcvw2 36173 . . 3 ([𝐴 / 𝑥]𝑡𝐶[𝐵 / 𝑦]𝑡𝐷)
54abbii 2805 . 2 {𝑡[𝐴 / 𝑥]𝑡𝐶} = {𝑡[𝐵 / 𝑦]𝑡𝐷}
6 df-csb 3909 . 2 𝐴 / 𝑥𝐶 = {𝑡[𝐴 / 𝑥]𝑡𝐶}
7 df-csb 3909 . 2 𝐵 / 𝑦𝐷 = {𝑡[𝐵 / 𝑦]𝑡𝐷}
85, 6, 73eqtr4i 2771 1 𝐴 / 𝑥𝐶 = 𝐵 / 𝑦𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = 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: (None)
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