Theorem difeq12 3276

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 3274	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
2		difeq2 3275	. 2 |- ( C = D -> ( B \ C ) = ( B \ D ) )
3	1, 2	sylan9eq 2249	1 |- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   \ cdif 3154

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-dif 3159

This theorem is referenced by:  resdif  5526