Theorem cbvcsbdavw2 36218

Description: Change bound variable of a proper substitution into a class. General version of cbvcsbdavw 36217. 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 2830	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvsbcdavw2 36216	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑡 ∈ 𝐷))
5	4	abbidv 2811	. 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶} = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷})
6		df-csb 3922	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶}
7		df-csb 3922	. 2 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4g 2805	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  [wsbc 3804  ⦋csb 3921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805  df-csb 3922

This theorem is referenced by: (None)