Theorem csbing 3247

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 2972	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷))
2		csbeq1 2972	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
3		csbeq1 2972	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
4	2, 3	ineq12d 3242	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
5	1, 4	eqeq12d 2127	. 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 2658	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 2999	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
8		nfcsb1v 2999	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐷
9	7, 8	nfin 3246	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
10		csbeq1a 2977	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
11		csbeq1a 2977	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷)
12	10, 11	ineq12d 3242	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷))
13	6, 9, 12	csbief 3008	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
14	5, 13	vtoclg 2715	1 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   = wceq 1312   ∈ wcel 1461  ⦋csb 2969   ∩ cin 3034

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-in 3041

This theorem is referenced by:  csbresg  4778