Theorem cbvsbcvw 3811

Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3813 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 30-Sep-2008.) Avoid ax-13 2369. (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		cbvsbcvw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cbvabv 2803	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
3	2	eleq2i 2823	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
4		df-sbc 3777	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
5		df-sbc 3777	. 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
6	3, 4, 5	3bitr4i 302	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2104  {cab 2707  [wsbc 3776

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-sbc 3777

This theorem is referenced by:  frpoins3xpg  8128  frpoins3xp3g  8129  fpwwe2cbv  10627  reuf1odnf  46113