Theorem cbvsbcvw2 36989

Description: Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw 3773. (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 2831	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
4	1, 3	eleq12i 2854	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∣ 𝜓})
5		df-sbc 3740	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
6		df-sbc 3740	. 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝐵 ∈ {𝑦 ∣ 𝜓})
7	4, 5, 6	3bitr4i 306	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2739  [wsbc 3739

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-sbc 3740

This theorem is used by:  cbvcsbvw2  36990