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Theorem difeq12 3057
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 3055 . 2  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3056 . 2  |-  ( C  =  D  ->  ( B  \  C )  =  ( B  \  D
) )
31, 2sylan9eq 2092 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C
)  =  ( B 
\  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    \ cdif 2914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-dif 2920
This theorem is referenced by:  resdif  5148
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