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Theorem cbvsbcvw 3756
Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3758 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by NM, 30-Sep-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypothesis
Ref Expression
cbvsbcvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvsbcvw ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem cbvsbcvw
StepHypRef Expression
1 nfv 1915 . 2 𝑦𝜑
2 nfv 1915 . 2 𝑥𝜓
3 cbvsbcvw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvsbcw 3755 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  [wsbc 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-sbc 3724
This theorem is referenced by:  fpwwe2cbv  10045  reuf1odnf  43650
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