Theorem cbvsbcdavw2 36495

Description: Change bound variable of a class substitution. General version of cbvsbcdavw 36494. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvsbcdavw2.1	⊢ (𝜑 → 𝐴 = 𝐵)
cbvsbcdavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvsbcdavw2	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑦]𝜒))

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

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

Proof of Theorem cbvsbcdavw2

Step	Hyp	Ref	Expression
1		cbvsbcdavw2.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		cbvsbcdavw2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
3	2	cbvabdavw 36493	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})
4	1, 3	eleq12d 2833	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∣ 𝜒}))
5		df-sbc 3724	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
6		df-sbc 3724	. 2 ⊢ ([𝐵 / 𝑦]𝜒 ↔ 𝐵 ∈ {𝑦 ∣ 𝜒})
7	4, 5, 6	3bitr4g 315	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑦]𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  [wsbc 3723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-sbc 3724

This theorem is referenced by:  cbvcsbdavw2  36497