Theorem cbvcsbdavw 36198

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 2819	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶))
3	2	cbvsbcdavw 36196	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐵 ↔ [𝐴 / 𝑦]𝑡 ∈ 𝐶))
4	3	abbidv 2800	. 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵} = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶})
5		df-csb 3873	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵}
6		df-csb 3873	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2794	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2712  [wsbc 3763  ⦋csb 3872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-sbc 3764  df-csb 3873

This theorem is referenced by:  cbvsumdavw  36218  cbvproddavw  36219  cbvsumdavw2  36234  cbvproddavw2  36235