Theorem indifcom 3453

Description: Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)

Assertion

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

Proof of Theorem indifcom

Step	Hyp	Ref	Expression
1		incom 3399	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	difeq1i 3321	. 2 |- ( ( A i^i B ) \ C ) = ( ( B i^i A ) \ C )
3		indif2 3451	. 2 |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )
4		indif2 3451	. 2 |- ( B i^i ( A \ C ) ) = ( ( B i^i A ) \ C )
5	2, 3, 4	3eqtr4i 2262	1 |- ( A i^i ( B \ C ) ) = ( B i^i ( A \ C ) )

Syntax hints:   = wceq 1397   \ cdif 3197   i^i cin 3199

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-dif 3202  df-in 3206

This theorem is referenced by: (None)