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Theorem symdifeq1 24435
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 3300 . . 3  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3301 . . 3  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2uneq12d 3343 . 2  |-  ( A  =  B  ->  (
( A  \  C
)  u.  ( C 
\  A ) )  =  ( ( B 
\  C )  u.  ( C  \  B
) ) )
4 df-symdif 24433 . 2  |-  ( A(++) C )  =  ( ( A  \  C
)  u.  ( C 
\  A ) )
5 df-symdif 24433 . 2  |-  ( B(++) C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
63, 4, 53eqtr4g 2353 1  |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    \ cdif 3162    u. cun 3163  (++)csymdif 24432
This theorem is referenced by:  symdifeq2  24436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-symdif 24433
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