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Theorem difeq12 3240
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
StepHypRef Expression
1 difeq1 3238 . 2  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3239 . 2  |-  ( C  =  D  ->  ( B  \  C )  =  ( B  \  D
) )
31, 2sylan9eq 2223 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C
)  =  ( B 
\  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    \ cdif 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-dif 3123
This theorem is referenced by:  resdif  5462
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