Theorem symdifeq1 3846

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

Step	Hyp	Ref	Expression
1		difeq1 3721	. . 3 |- ( A = B -> ( A \ C ) = ( B \ C ) )
2		difeq2 3722	. . 3 |- ( A = B -> ( C \ A ) = ( C \ B ) )
3	1, 2	uneq12d 3768	. 2 |- ( A = B -> ( ( A \ C ) u. ( C \ A ) ) = ( ( B \ C ) u. ( C \ B ) ) )
4		df-symdif 3844	. 2 |- ( A /_\ C ) = ( ( A \ C ) u. ( C \ A ) )
5		df-symdif 3844	. 2 |- ( B /_\ C ) = ( ( B \ C ) u. ( C \ B ) )
6	3, 4, 5	3eqtr4g 2681	1 |- ( A = B -> ( A /_\ C ) = ( B /_\ C ) )

Syntax hints:   -> wi 4   = wceq 1483   \ cdif 3571   u. cun 3572   /_\ csymdif 3843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-symdif 3844

This theorem is referenced by:  symdifeq2  3847