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Theorem cbvsbcvw 3821
Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3823 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 30-Sep-2008.) Avoid ax-13 2376. (Revised by GG, 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 cbvsbcvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21cbvabv 2811 . . 3 {𝑥𝜑} = {𝑦𝜓}
32eleq2i 2832 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑦𝜓})
4 df-sbc 3788 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
5 df-sbc 3788 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
63, 4, 53bitr4i 303 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2107  {cab 2713  [wsbc 3787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-sbc 3788
This theorem is referenced by:  cbvcsbv  3910  frpoins3xpg  8166  frpoins3xp3g  8167  fpwwe2cbv  10671  reuf1odnf  47124
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