Theorem cbvcsbdavw 36293

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 2817	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶))
3	2	cbvsbcdavw 36291	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐵 ↔ [𝐴 / 𝑦]𝑡 ∈ 𝐶))
4	3	abbidv 2797	. 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵} = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶})
5		df-csb 3846	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵}
6		df-csb 3846	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2791	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  [wsbc 3736  ⦋csb 3845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3737  df-csb 3846

This theorem is referenced by:  cbvsumdavw  36313  cbvproddavw  36314  cbvsumdavw2  36329  cbvproddavw2  36330