Theorem difeq12 3086

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 3084	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
2		difeq2 3085	. 2 |- ( C = D -> ( B \ C ) = ( B \ D ) )
3	1, 2	sylan9eq 2134	1 |- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   \ cdif 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-dif 2976

This theorem is referenced by:  resdif  5179