Theorem indifcom 3427

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 3373	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	difeq1i 3295	. 2 |- ( ( A i^i B ) \ C ) = ( ( B i^i A ) \ C )
3		indif2 3425	. 2 |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )
4		indif2 3425	. 2 |- ( B i^i ( A \ C ) ) = ( ( B i^i A ) \ C )
5	2, 3, 4	3eqtr4i 2238	1 |- ( A i^i ( B \ C ) ) = ( B i^i ( A \ C ) )

Syntax hints:   = wceq 1373   \ cdif 3171   i^i cin 3173

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-dif 3176  df-in 3180

This theorem is referenced by: (None)