Theorem indif1 3380

Description: Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
indif1	|- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C )

Proof of Theorem indif1

Step	Hyp	Ref	Expression
1		indif2 3379	. 2 |- ( B i^i ( A \ C ) ) = ( ( B i^i A ) \ C )
2		incom 3327	. 2 |- ( B i^i ( A \ C ) ) = ( ( A \ C ) i^i B )
3		incom 3327	. . 3 |- ( B i^i A ) = ( A i^i B )
4	3	difeq1i 3249	. 2 |- ( ( B i^i A ) \ C ) = ( ( A i^i B ) \ C )
5	1, 2, 4	3eqtr3i 2206	1 |- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C )

Syntax hints:   = wceq 1353   \ cdif 3126   i^i cin 3128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135

This theorem is referenced by:  resdifcom  4924  resdmdfsn  4949