Theorem difeq12 3286

Description: Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
difeq12	|- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) )

Proof of Theorem difeq12

Step	Hyp	Ref	Expression
1		difeq1 3284	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
2		difeq2 3285	. 2 |- ( C = D -> ( B \ C ) = ( B \ D ) )
3	1, 2	sylan9eq 2258	1 |- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   \ cdif 3163

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-dif 3168

This theorem is referenced by:  resdif  5544