Theorem csbing 3463

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 3140	. . 3 ⊢ (y = A → [y / x](C ∩ D) = [A / x](C ∩ D))
2		csbeq1 3140	. . . 4 ⊢ (y = A → [y / x]C = [A / x]C)
3		csbeq1 3140	. . . 4 ⊢ (y = A → [y / x]D = [A / x]D)
4	2, 3	ineq12d 3459	. . 3 ⊢ (y = A → ([y / x]C ∩ [y / x]D) = ([A / x]C ∩ [A / x]D))
5	1, 4	eqeq12d 2367	. 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 2863	. . 3 ⊢ y ∈ V
7		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]C
8		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]D
9	7, 8	nfin 3231	. . 3 ⊢ Ⅎx([y / x]C ∩ [y / x]D)
10		csbeq1a 3145	. . . 4 ⊢ (x = y → C = [y / x]C)
11		csbeq1a 3145	. . . 4 ⊢ (x = y → D = [y / x]D)
12	10, 11	ineq12d 3459	. . 3 ⊢ (x = y → (C ∩ D) = ([y / x]C ∩ [y / x]D))
13	6, 9, 12	csbief 3178	. 2 ⊢ [y / x](C ∩ D) = ([y / x]C ∩ [y / x]D)
14	5, 13	vtoclg 2915	1 ⊢ (A ∈ B → [A / x](C ∩ D) = ([A / x]C ∩ [A / x]D))

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  [csb 3137   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by:  csbresg  4977