Theorem xorcom 1383

Description: \/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)

Assertion

Ref	Expression
xorcom	|- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )

Proof of Theorem xorcom

Step	Hyp	Ref	Expression
1		orcom 723	. . 3 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
2		ancom 264	. . . 4 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3	2	notbii 663	. . 3 |- ( -. ( ph /\ ps ) <-> -. ( ps /\ ph ) )
4	1, 3	anbi12i 457	. 2 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ( ps \/ ph ) /\ -. ( ps /\ ph ) ) )
5		df-xor 1371	. 2 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
6		df-xor 1371	. 2 |- ( ( ps \/_ ph ) <-> ( ( ps \/ ph ) /\ -. ( ps /\ ph ) ) )
7	4, 5, 6	3bitr4i 211	1 |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   \/ wo 703   \/_ wxo 1370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-xor 1371

This theorem is referenced by:  rpnegap  9643