Theorem iinin1 4122

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4105 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 4121	. 2 |- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
2		incom 3493	. . . 4 |- ( C i^i B ) = ( B i^i C )
3	2	a1i 11	. . 3 |- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iineq2i 4072	. 2 |- |^|_ x e. A ( C i^i B ) = |^|_ x e. A ( B i^i C )
5		incom 3493	. 2 |- ( |^|_ x e. A C i^i B ) = ( B i^i |^|_ x e. A C )
6	1, 4, 5	3eqtr4g 2461	1 |- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   =/= wne 2567   i^i cin 3279   (/)c0 3588   |^|_ciin 4054

This theorem is referenced by:  firest  13615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589  df-iin 4056