Theorem cbvcsbv 3911

Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
cbvcsbv.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

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

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

Proof of Theorem cbvcsbv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsbv.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
2	1	eleq2d 2827	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
3	2	cbvsbcvw 3822	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐶)
4	3	abbii 2809	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐶}
5		df-csb 3900	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
6		df-csb 3900	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐶}
7	4, 5, 6	3eqtr4i 2775	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {cab 2714  [wsbc 3788  ⦋csb 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789  df-csb 3900

This theorem is referenced by:  cbvsumv  15732  cbvprodv  15950  pmatcollpw3lem  22789  precsexlemcbv  28230  cbvprodvw2  36248  poimirlem27  37654  cdleme40v  40471