Theorem cadcoma 1551

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

Assertion

Ref	Expression
cadcoma	|- (cadd ( ph , ps , ch ) <-> cadd ( ps , ph , ch ) )

Proof of Theorem cadcoma

Step	Hyp	Ref	Expression
1		ancom 466	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
2		xorcom 1467	. . . 4 |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )
3	2	anbi2i 730	. . 3 |- ( ( ch /\ ( ph \/_ ps ) ) <-> ( ch /\ ( ps \/_ ph ) ) )
4	1, 3	orbi12i 543	. 2 |- ( ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) <-> ( ( ps /\ ph ) \/ ( ch /\ ( ps \/_ ph ) ) ) )
5		df-cad 1546	. 2 |- (cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) )
6		df-cad 1546	. 2 |- (cadd ( ps , ph , ch ) <-> ( ( ps /\ ph ) \/ ( ch /\ ( ps \/_ ph ) ) ) )
7	4, 5, 6	3bitr4i 292	1 |- (cadd ( ph , ps , ch ) <-> cadd ( ps , ph , ch ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   \/_ wxo 1464  caddwcad 1545

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-or 385  df-an 386  df-xor 1465  df-cad 1546

This theorem is referenced by:  cadrot  1553  sadcom  15185