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Theorem symdifeq1 27987
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 3567 . . 3  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3568 . . 3  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2uneq12d 3611 . 2  |-  ( A  =  B  ->  (
( A  \  C
)  u.  ( C 
\  A ) )  =  ( ( B 
\  C )  u.  ( C  \  B
) ) )
4 df-symdif 27985 . 2  |-  ( A(++) C )  =  ( ( A  \  C
)  u.  ( C 
\  A ) )
5 df-symdif 27985 . 2  |-  ( B(++) C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
63, 4, 53eqtr4g 2517 1  |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    \ cdif 3425    u. cun 3426  (++)csymdif 27984
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 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-symdif 27985
This theorem is referenced by:  symdifeq2  27988
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