Theorem iinin1 3974

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3957 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
iinin1	|- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iinin1

Step	Hyp	Ref	Expression
1		iinin2 3973	. 2 |- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
2		incom 3362	. . . 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	iineq2i 3925	. 2 |- |^|_ x e. A ( C i^i B ) = |^|_ x e. A ( B i^i C )
5		incom 3362	. 2 |- ( |^|_ x e. A C i^i B ) = ( B i^i |^|_ x e. A C )
6	1, 4, 5	3eqtr4g 2341	1 |- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1623   e. wcel 1685   =/= wne 2447   i^i cin 3152   (/)c0 3456   |^|_ciin 3907

This theorem is referenced by:  firest  13333

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-v 2791  df-dif 3156  df-in 3160  df-nul 3457  df-iin 3909