Theorem hadass 1536

Description: Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
hadass	|- (hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )

Proof of Theorem hadass

Step	Hyp	Ref	Expression
1		df-had 1533	. 2 |- (hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) )
2		xorass 1468	. 2 |- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )
3	1, 2	bitri 264	1 |- (hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ 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:  hadcomb  1539