Theorem hadcoma 1538

Description: Commutative law for the adder sum. (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 1467	. . 3 |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )
2		biid 251	. . 3 |- ( ch <-> ch )
3	1, 2	xorbi12i 1477	. 2 |- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ( ps \/_ ph ) \/_ ch ) )
4		df-had 1533	. 2 |- (hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) )
5		df-had 1533	. 2 |- (hadd ( ps , ph , ch ) <-> ( ( ps \/_ ph ) \/_ ch ) )
6	3, 4, 5	3bitr4i 292	1 |- (hadd ( ph , ps , ch ) <-> hadd ( ps , ph , ch ) )

Syntax hints:   <-> wb 196   \/_ wxo 1464  haddwhad 1532

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 197  df-xor 1465  df-had 1533

This theorem is referenced by:  hadrot  1540  sadcom  15185