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Theorem symdifeq1 28897
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 3608 . . 3  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3609 . . 3  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2uneq12d 3652 . 2  |-  ( A  =  B  ->  (
( A  \  C
)  u.  ( C 
\  A ) )  =  ( ( B 
\  C )  u.  ( C  \  B
) ) )
4 df-symdif 28895 . 2  |-  ( A(++) C )  =  ( ( A  \  C
)  u.  ( C 
\  A ) )
5 df-symdif 28895 . 2  |-  ( B(++) C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
63, 4, 53eqtr4g 2526 1  |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    \ cdif 3466    u. cun 3467  (++)csymdif 28894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-symdif 28895
This theorem is referenced by:  symdifeq2  28898
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