Theorem xorcom 1388

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 728	. . 3 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
2		ancom 266	. . . 4 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3	2	notbii 668	. . 3 |- ( -. ( ph /\ ps ) <-> -. ( ps /\ ph ) )
4	1, 3	anbi12i 460	. 2 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ( ps \/ ph ) /\ -. ( ps /\ ph ) ) )
5		df-xor 1376	. 2 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
6		df-xor 1376	. 2 |- ( ( ps \/_ ph ) <-> ( ( ps \/ ph ) /\ -. ( ps /\ ph ) ) )
7	4, 5, 6	3bitr4i 212	1 |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 708   \/_ wxo 1375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-xor 1376

This theorem is referenced by:  rpnegap  9681