Theorem indif1 3394

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 3393	. 2 |- ( B i^i ( A \ C ) ) = ( ( B i^i A ) \ C )
2		incom 3341	. 2 |- ( B i^i ( A \ C ) ) = ( ( A \ C ) i^i B )
3		incom 3341	. . 3 |- ( B i^i A ) = ( A i^i B )
4	3	difeq1i 3263	. 2 |- ( ( B i^i A ) \ C ) = ( ( A i^i B ) \ C )
5	1, 2, 4	3eqtr3i 2217	1 |- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C )

Syntax hints:   = wceq 1363   \ cdif 3140   i^i cin 3142

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rab 2476  df-v 2753  df-dif 3145  df-in 3149

This theorem is referenced by:  resdifcom  4939  resdmdfsn  4964