Theorem csbing 3391

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Assertion

Ref	Expression
csbing	⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Proof of Theorem csbing

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3107	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷))
2		csbeq1 3107	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
3		csbeq1 3107	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
4	2, 3	ineq12d 3386	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
5	1, 4	eqeq12d 2224	. 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷) ↔ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)))
6		vex 2782	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3137	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
8		nfcsb1v 3137	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐷
9	7, 8	nfin 3390	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
10		csbeq1a 3113	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
11		csbeq1a 3113	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷)
12	10, 11	ineq12d 3386	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷))
13	6, 9, 12	csbief 3149	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
14	5, 13	vtoclg 2841	1 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   = wceq 1375   ∈ wcel 2180  ⦋csb 3104   ∩ cin 3176

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-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-in 3183

This theorem is referenced by:  csbresg  4984