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Theorem difeq12 3603
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 3601 . 2  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3602 . 2  |-  ( C  =  D  ->  ( B  \  C )  =  ( B  \  D
) )
31, 2sylan9eq 2515 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C
)  =  ( B 
\  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    \ cdif 3458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-dif 3464
This theorem is referenced by:  resdif  5818
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