Theorem cbvcsbdavw2 36829

Description: Change bound variable of a proper substitution into a class. General version of cbvcsbdavw 36828. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvcsbdavw2.1	⊢ (𝜑 → 𝐴 = 𝐵)
cbvcsbdavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cbvcsbdavw2	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷)

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

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

Proof of Theorem cbvcsbdavw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsbdavw2.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
2		cbvcsbdavw2.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
3	2	eleq2d 2851	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvsbcdavw2 36827	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑡 ∈ 𝐷))
5	4	abbidv 2831	. 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶} = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷})
6		df-csb 3855	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶}
7		df-csb 3855	. 2 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4g 2825	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷)

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: (None)