Theorem cbvsbcvw2 36443

Description: Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw 3776. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
cbvsbcvw2.1	⊢ 𝐴 = 𝐵
cbvsbcvw2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvsbcvw2	⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem cbvsbcvw2

Step	Hyp	Ref	Expression
1		cbvsbcvw2.1	. . 3 ⊢ 𝐴 = 𝐵
2		cbvsbcvw2.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	cbvabv 2807	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
4	1, 3	eleq12i 2830	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∣ 𝜓})
5		df-sbc 3743	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
6		df-sbc 3743	. 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝐵 ∈ {𝑦 ∣ 𝜓})
7	4, 5, 6	3bitr4i 303	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2715  [wsbc 3742

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sbc 3743

This theorem is referenced by:  cbvcsbvw2  36444