Theorem indif2 3325

Description: Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)

Assertion

Ref	Expression
indif2	|- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )

Proof of Theorem indif2

Step	Hyp	Ref	Expression
1		inass 3291	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( A i^i ( B i^i ( _V \ C ) ) )
2		invdif 3323	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif 3323	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
4	3	ineq2i 3279	. 2 |- ( A i^i ( B i^i ( _V \ C ) ) ) = ( A i^i ( B \ C ) )
5	1, 2, 4	3eqtr3ri 2170	1 |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )

Syntax hints:   = wceq 1332   _Vcvv 2689   \ cdif 3073   i^i cin 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082

This theorem is referenced by:  indif1  3326  indifcom  3327  difopn  12316