Theorem elsymdifxor 3850

Description: Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.)

Assertion

Ref	Expression
elsymdifxor	|- ( A e. ( B /_\ C ) <-> ( A e. B \/_ A e. C ) )

Proof of Theorem elsymdifxor

Step	Hyp	Ref	Expression
1		xnor 1466	. . 3 |- ( ( A e. B <-> A e. C ) <-> -. ( A e. B \/_ A e. C ) )
2	1	notbii 310	. 2 |- ( -. ( A e. B <-> A e. C ) <-> -. -. ( A e. B \/_ A e. C ) )
3		elsymdif 3849	. 2 |- ( A e. ( B /_\ C ) <-> -. ( A e. B <-> A e. C ) )
4		notnotb 304	. 2 |- ( ( A e. B \/_ A e. C ) <-> -. -. ( A e. B \/_ A e. C ) )
5	2, 3, 4	3bitr4i 292	1 |- ( A e. ( B /_\ C ) <-> ( A e. B \/_ A e. C ) )

Syntax hints:   -. wn 3   <-> wb 196   \/_ wxo 1464   e. wcel 1990   /_\ 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-xor 1465  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-v 3202  df-dif 3577  df-un 3579  df-symdif 3844

This theorem is referenced by:  dfsymdif2  3851