Theorem hadcomb 1389

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

Assertion

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

Proof of Theorem hadcomb

Step	Hyp	Ref	Expression
1		biid 227	. . 3 |- (ph <-> ph)
2		xorcom 1307	. . 3 |- ((ps \/_ ch) <-> (ch \/_ ps))
3	1, 2	xorbi12i 1314	. 2 |- ((ph \/_ (ps \/_ ch)) <-> (ph \/_ (ch \/_ ps)))
4		hadass 1386	. 2 |- (hadd(ph, ps, ch) <-> (ph \/_ (ps \/_ ch)))
5		hadass 1386	. 2 |- (hadd(ph, ch, ps) <-> (ph \/_ (ch \/_ ps)))
6	3, 4, 5	3bitr4i 268	1 |- (hadd(ph, ps, ch) <-> hadd(ph, ch, ps))

This proof depends on syntax axioms:   <-> wb 176   \/_ wxo 1304  haddwhad 1378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-xor 1305  df-had 1380

This theorem is used by:  hadrot  1390