Theorem cbvsbcvw 3780

Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3782 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 30-Sep-2008.) Avoid ax-13 2406. (Revised by GG, 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 2835	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
3	2	eleq2i 2857	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
4		df-sbc 3747	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
5		df-sbc 3747	. 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
6	3, 4, 5	3bitr4i 306	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146  {cab 2743  [wsbc 3746

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-sbc 3747

This theorem is used by:  cbvcsbv  3866  frpoins3xpg  8142  frpoins3xp3g  8143  fpwwe2cbv  10630  reuf1odnf  47902