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Theorem symdifeq1 25657
Description: Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdifeq1  |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )

Proof of Theorem symdifeq1
StepHypRef Expression
1 difeq1 3450 . . 3  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3451 . . 3  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2uneq12d 3494 . 2  |-  ( A  =  B  ->  (
( A  \  C
)  u.  ( C 
\  A ) )  =  ( ( B 
\  C )  u.  ( C  \  B
) ) )
4 df-symdif 25655 . 2  |-  ( A(++) C )  =  ( ( A  \  C
)  u.  ( C 
\  A ) )
5 df-symdif 25655 . 2  |-  ( B(++) C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
63, 4, 53eqtr4g 2492 1  |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    \ cdif 3309    u. cun 3310  (++)csymdif 25654
This theorem is referenced by:  symdifeq2  25658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-symdif 25655
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