Theorem cadcomb 1552

Description: Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Assertion

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

Proof of Theorem cadcomb

Step	Hyp	Ref	Expression
1		cadan 1548	. . 3 |- (cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) )
2		3ancoma 1045	. . 3 |- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ph \/ ch ) /\ ( ph \/ ps ) /\ ( ps \/ ch ) ) )
3		orcom 402	. . . 4 |- ( ( ps \/ ch ) <-> ( ch \/ ps ) )
4	3	3anbi3i 1255	. . 3 |- ( ( ( ph \/ ch ) /\ ( ph \/ ps ) /\ ( ps \/ ch ) ) <-> ( ( ph \/ ch ) /\ ( ph \/ ps ) /\ ( ch \/ ps ) ) )
5	1, 2, 4	3bitri 286	. 2 |- (cadd ( ph , ps , ch ) <-> ( ( ph \/ ch ) /\ ( ph \/ ps ) /\ ( ch \/ ps ) ) )
6		cadan 1548	. 2 |- (cadd ( ph , ch , ps ) <-> ( ( ph \/ ch ) /\ ( ph \/ ps ) /\ ( ch \/ ps ) ) )
7	5, 6	bitr4i 267	1 |- (cadd ( ph , ps , ch ) <-> cadd ( ph , ch , ps ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ w3a 1037  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-3or 1038  df-3an 1039  df-xor 1465  df-cad 1546

This theorem is referenced by:  cadrot  1553