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Theorem symdifeq1 3717
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 3601 . . 3  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3602 . . 3  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2uneq12d 3645 . 2  |-  ( A  =  B  ->  (
( A  \  C
)  u.  ( C 
\  A ) )  =  ( ( B 
\  C )  u.  ( C  \  B
) ) )
4 df-symdif 3715 . 2  |-  ( A  /_\  C )  =  ( ( A  \  C
)  u.  ( C 
\  A ) )
5 df-symdif 3715 . 2  |-  ( B  /_\  C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
63, 4, 53eqtr4g 2520 1  |-  ( A  =  B  ->  ( A  /_\  C )  =  ( B  /_\  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    \ cdif 3458    u. cun 3459    /_\ csymdif 3714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-symdif 3715
This theorem is referenced by:  symdifeq2  3718
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