Theorem indif1 3378

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 3377	. 2 |- ( B i^i ( A \ C ) ) = ( ( B i^i A ) \ C )
2		incom 3325	. 2 |- ( B i^i ( A \ C ) ) = ( ( A \ C ) i^i B )
3		incom 3325	. . 3 |- ( B i^i A ) = ( A i^i B )
4	3	difeq1i 3247	. 2 |- ( ( B i^i A ) \ C ) = ( ( A i^i B ) \ C )
5	1, 2, 4	3eqtr3i 2204	1 |- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C )

Syntax hints:   = wceq 1353   \ cdif 3124   i^i cin 3126

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-v 2737  df-dif 3129  df-in 3133

This theorem is referenced by:  resdifcom  4918  resdmdfsn  4943