Theorem cbvcsbvw2 37211

Description: Change bound variable of a proper substitution into a class using implicit substitution. General version of cbvcsbv 3863. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
cbvcsbvw2.1	⊢ 𝐴 = 𝐵
cbvcsbvw2.2	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)

Assertion

Ref	Expression
cbvcsbvw2	⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷

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

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

Proof of Theorem cbvcsbvw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsbvw2.1	. . . 4 ⊢ 𝐴 = 𝐵
2		cbvcsbvw2.2	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
3	2	eleq2d 2850	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvsbcvw2 37210	. . 3 ⊢ ([𝐴 / 𝑥]𝑡 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑡 ∈ 𝐷)
5	4	abbii 2830	. 2 ⊢ {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶} = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷}
6		df-csb 3852	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶}
7		df-csb 3852	. 2 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2796	1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741  [wsbc 3742  ⦋csb 3851

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743  df-csb 3852

This theorem is used by: (None)