Theorem iniin2 39310

Description: Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
iniin2	|- ( A =/= (/) -> ( B i^i |^|_ x e. A C ) = |^|_ x e. A ( B i^i C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iniin2

Step	Hyp	Ref	Expression
1		iinin2 4590	. 2 |- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
2	1	eqcomd 2628	1 |- ( A =/= (/) -> ( B i^i |^|_ x e. A C ) = |^|_ x e. A ( B i^i C ) )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   i^i cin 3573   (/)c0 3915   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-iin 4523

This theorem is referenced by: (None)