Theorem cbvcsbdavw2 36660

Description: Change bound variable of a proper substitution into a class. General version of cbvcsbdavw 36659. 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 2855	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvsbcdavw2 36658	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑡 ∈ 𝐷))
5	4	abbidv 2835	. 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶} = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷})
6		df-csb 3862	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶}
7		df-csb 3862	. 2 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4g 2829	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  [wsbc 3753  ⦋csb 3861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sbc 3754  df-csb 3862

This theorem is referenced by: (None)