Theorem iunin1 4032

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4020 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
iunin1	|- U_x e. A (C i^i B) = (U_x e. A C i^i B)

Distinct variable group:   x,B

Allowed substitution hints:   A(x)   C(x)

Proof of Theorem iunin1

Step	Hyp	Ref	Expression
1		iunin2 4031	. 2 |- U_x e. A (B i^i C) = (B i^i U_x e. A C)
2		incom 3449	. . . 4 |- (C i^i B) = (B i^i C)
3	2	a1i 10	. . 3 |- (x e. A -> (C i^i B) = (B i^i C))
4	3	iuneq2i 3988	. 2 |- U_x e. A (C i^i B) = U_x e. A (B i^i C)
5		incom 3449	. 2 |- (U_x e. A C i^i B) = (B i^i U_x e. A C)
6	1, 4, 5	3eqtr4i 2383	1 |- U_x e. A (C i^i B) = (U_x e. A C i^i B)

Syntax hints:   = wceq 1642   e. wcel 1710   i^i cin 3209  U_ciun 3970

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-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-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by:  2iunin  4035