Theorem cbvsbcvw 3801

Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3803 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 30-Sep-2008.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypothesis

Ref	Expression
cbvsbcvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

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

Step	Hyp	Ref	Expression
1		nfv 1914	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvsbcvw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvsbcw 3800	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4   ↔ wb 208  [wsbc 3768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-sbc 3769

This theorem is referenced by:  fpwwe2cbv  10045  reuf1odnf  43380