Theorem iinin1 3947

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3930 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 3946	. 2 |- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
2		incom 3336	. . . 4 |- ( C i^i B ) = ( B i^i C )
3	2	a1i 12	. . 3 |- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iineq2i 3898	. 2 |- |^|_ x e. A ( C i^i B ) = |^|_ x e. A ( B i^i C )
5		incom 3336	. 2 |- ( |^|_ x e. A C i^i B ) = ( B i^i |^|_ x e. A C )
6	1, 4, 5	3eqtr4g 2315	1 |- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )

Syntax hints:   -> wi 6   = wceq 1619   e. wcel 1621   =/= wne 2421   i^i cin 3126   (/)c0 3430   |^|_ciin 3880

This theorem is referenced by:  firest  13300

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-v 2765  df-dif 3130  df-in 3134  df-nul 3431  df-iin 3882