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Theorem symdifeq1 25388
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 3401 . . 3  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3402 . . 3  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2uneq12d 3445 . 2  |-  ( A  =  B  ->  (
( A  \  C
)  u.  ( C 
\  A ) )  =  ( ( B 
\  C )  u.  ( C  \  B
) ) )
4 df-symdif 25386 . 2  |-  ( A(++) C )  =  ( ( A  \  C
)  u.  ( C 
\  A ) )
5 df-symdif 25386 . 2  |-  ( B(++) C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
63, 4, 53eqtr4g 2444 1  |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    \ cdif 3260    u. cun 3261  (++)csymdif 25385
This theorem is referenced by:  symdifeq2  25389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-symdif 25386
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