Theorem cad1 1555

Description: If one input is true, then the adder carry is true exactly when at least one of the other two inputs is true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 19-Jun-2020.)

Assertion

Ref	Expression
cad1	|- ( ch -> (cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) )

Proof of Theorem cad1

Step	Hyp	Ref	Expression
1		olc 399	. . . 4 |- ( ch -> ( ph \/ ch ) )
2		olc 399	. . . 4 |- ( ch -> ( ps \/ ch ) )
3	1, 2	jca 554	. . 3 |- ( ch -> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )
4	3	biantrud 528	. 2 |- ( ch -> ( ( ph \/ ps ) <-> ( ( ph \/ ps ) /\ ( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) ) )
5		cadan 1548	. . 3 |- (cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) )
6		3anass 1042	. . 3 |- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ph \/ ps ) /\ ( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) )
7	5, 6	bitri 264	. 2 |- (cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) )
8	4, 7	syl6rbbr 279	1 |- ( ch -> (cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ 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:  cadifp  1557  sadadd2lem2  15172  sadcaddlem  15179