Theorem csbeq2d 3119

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 2276	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
4	1, 3	sbcbid 3057	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2324	. 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6		df-csb 3095	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
7		df-csb 3095	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2264	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1373  Ⅎwnf 1484   ∈ wcel 2177  {cab 2192  [wsbc 2999  ⦋csb 3094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-sbc 3000  df-csb 3095

This theorem is referenced by:  csbeq2dv  3120