Theorem hadcoma 1388

Description: Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
hadcoma	|- (hadd(ph, ps, ch) <-> hadd(ps, ph, ch))

Proof of Theorem hadcoma

Step	Hyp	Ref	Expression
1		xorcom 1307	. . 3 |- ((ph \/_ ps) <-> (ps \/_ ph))
2		biid 227	. . 3 |- (ch <-> ch)
3	1, 2	xorbi12i 1314	. 2 |- (((ph \/_ ps) \/_ ch) <-> ((ps \/_ ph) \/_ ch))
4		df-had 1380	. 2 |- (hadd(ph, ps, ch) <-> ((ph \/_ ps) \/_ ch))
5		df-had 1380	. 2 |- (hadd(ps, ph, ch) <-> ((ps \/_ ph) \/_ ch))
6	3, 4, 5	3bitr4i 268	1 |- (hadd(ph, ps, ch) <-> hadd(ps, ph, ch))

Syntax hints:   <-> wb 176   \/_ wxo 1304  haddwhad 1378

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  df-had 1380

This theorem is referenced by:  hadrot  1390