Theorem difeq12 3342

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 3340	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
2		difeq2 3341	. 2 |- ( C = D -> ( B \ C ) = ( B \ D ) )
3	1, 2	sylan9eq 2291	1 |- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   \ cdif 3217

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-dif 3222

This theorem is referenced by:  resdif  5656