Theorem xorcom 1307

Description: \/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

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

Proof of Theorem xorcom

Step	Hyp	Ref	Expression
1		bicom 191	. . 3 |- ((ph <-> ps) <-> (ps <-> ph))
2	1	notbii 287	. 2 |- (-. (ph <-> ps) <-> -. (ps <-> ph))
3		df-xor 1305	. 2 |- ((ph \/_ ps) <-> -. (ph <-> ps))
4		df-xor 1305	. 2 |- ((ps \/_ ph) <-> -. (ps <-> ph))
5	2, 3, 4	3bitr4i 268	1 |- ((ph \/_ ps) <-> (ps \/_ ph))

Syntax hints:  -. wn 3   <-> wb 176   \/_ wxo 1304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-xor 1305

This theorem is referenced by:  xorneg2  1312  hadcoma  1388  hadcomb  1389  cadcoma  1395