Theorem cbvcsbdavw 36487

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

Hypothesis

Ref	Expression
cbvcsbdavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvcsbdavw	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvcsbdavw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsbdavw.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
2	1	eleq2d 2825	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶))
3	2	cbvsbcdavw 36485	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐵 ↔ [𝐴 / 𝑦]𝑡 ∈ 𝐶))
4	3	abbidv 2805	. 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵} = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶})
5		df-csb 3832	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵}
6		df-csb 3832	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2799	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  [wsbc 3723  ⦋csb 3831

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  df-csb 3832

This theorem is referenced by:  cbvsumdavw  36507  cbvproddavw  36508  cbvsumdavw2  36523  cbvproddavw2  36524