Theorem cbvsbcdavw 36657

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

Hypothesis

Ref	Expression
cbvsbcdavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

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

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

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

Proof of Theorem cbvsbcdavw

Step	Hyp	Ref	Expression
1		cbvsbcdavw.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2	1	cbvabdavw 36656	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})
3	2	eleq2d 2855	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑦 ∣ 𝜒}))
4		df-sbc 3754	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
5		df-sbc 3754	. 2 ⊢ ([𝐴 / 𝑦]𝜒 ↔ 𝐴 ∈ {𝑦 ∣ 𝜒})
6	3, 4, 5	3bitr4g 317	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑦]𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  {cab 2747  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sbc 3754

This theorem is referenced by:  cbvcsbdavw  36659