Theorem cbvcsbdavw 36828

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 2851	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶))
3	2	cbvsbcdavw 36826	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐵 ↔ [𝐴 / 𝑦]𝑡 ∈ 𝐶))
4	3	abbidv 2831	. 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵} = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶})
5		df-csb 3855	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐵}
6		df-csb 3855	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑦]𝑡 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2825	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743  [wsbc 3746  ⦋csb 3854

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-sbc 3747  df-csb 3855

This theorem is used by:  cbvsumdavw  36848  cbvproddavw  36849  cbvsumdavw2  36864  cbvproddavw2  36865