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Theorem difeq12 3553
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 3551 . 2  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3552 . 2  |-  ( C  =  D  ->  ( B  \  C )  =  ( B  \  D
) )
31, 2sylan9eq 2510 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C
)  =  ( B 
\  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    \ cdif 3409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ral 2797  df-rab 2801  df-dif 3415
This theorem is referenced by:  resdif  5745
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