Theorem csbeq2d 3835

Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
csbeq2d.1	⊢ Ⅎ𝑥𝜑
csbeq2d.2	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbeq2d	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbeq2d

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq2d.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		csbeq2d.2	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝐶)
3	2	eleq2d 2825	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
4	1, 3	sbcbid 3770	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2809	. 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6		df-csb 3830	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
7		df-csb 3830	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2805	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1543  Ⅎwnf 1791   ∈ wcel 2112  {cab 2716  [wsbc 3712  ⦋csb 3829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2177  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-sbc 3713  df-csb 3830

This theorem is referenced by:  csbeq2dv  3836  poimirlem26  35709