Theorem iinin1

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

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

Proof

Step	Hyp	Ref	Expression
1		iinin2	|- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
2		incom	|- ( C i^i B ) = ( B i^i C )
3	2	a1i	|- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iineq2i	|- |^|_ x e. A ( C i^i B ) = |^|_ x e. A ( B i^i C )
5		incom	|- ( |^|_ x e. A C i^i B ) = ( B i^i |^|_ x e. A C )
6	1 4 5	3eqtr4g	|- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )