Theorem cad0 1556

Description: If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016.)

Assertion

Ref	Expression
cad0	|- ( -. ch -> (cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) )

Proof of Theorem cad0

Step	Hyp	Ref	Expression
1		df-cad 1546	. 2 |- (cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) )
2		idd 24	. . . 4 |- ( -. ch -> ( ( ph /\ ps ) -> ( ph /\ ps ) ) )
3		pm2.21 120	. . . . 5 |- ( -. ch -> ( ch -> ( ph /\ ps ) ) )
4	3	adantrd 484	. . . 4 |- ( -. ch -> ( ( ch /\ ( ph \/_ ps ) ) -> ( ph /\ ps ) ) )
5	2, 4	jaod 395	. . 3 |- ( -. ch -> ( ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) -> ( ph /\ ps ) ) )
6		orc 400	. . 3 |- ( ( ph /\ ps ) -> ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) )
7	5, 6	impbid1 215	. 2 |- ( -. ch -> ( ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) <-> ( ph /\ ps ) ) )
8	1, 7	syl5bb 272	1 |- ( -. ch -> (cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> 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-cad 1546

This theorem is referenced by:  cadifp  1557  sadadd2lem2  15172  sadcaddlem  15179  saddisjlem  15186