Theorem cadbi123d 1549

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

Hypotheses

Ref	Expression
cadbid.1	|- ( ph -> ( ps <-> ch ) )
cadbid.2	|- ( ph -> ( th <-> ta ) )
cadbid.3	|- ( ph -> ( et <-> ze ) )

Assertion

Ref	Expression
cadbi123d	|- ( ph -> (cadd ( ps , th , et ) <-> cadd ( ch , ta , ze ) ) )

Proof of Theorem cadbi123d

Step	Hyp	Ref	Expression
1		cadbid.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
2		cadbid.2	. . . 4 |- ( ph -> ( th <-> ta ) )
3	1, 2	anbi12d 747	. . 3 |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ ta ) ) )
4		cadbid.3	. . . 4 |- ( ph -> ( et <-> ze ) )
5	1, 2	xorbi12d 1478	. . . 4 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )
6	4, 5	anbi12d 747	. . 3 |- ( ph -> ( ( et /\ ( ps \/_ th ) ) <-> ( ze /\ ( ch \/_ ta ) ) ) )
7	3, 6	orbi12d 746	. 2 |- ( ph -> ( ( ( ps /\ th ) \/ ( et /\ ( ps \/_ th ) ) ) <-> ( ( ch /\ ta ) \/ ( ze /\ ( ch \/_ ta ) ) ) ) )
8		df-cad 1546	. 2 |- (cadd ( ps , th , et ) <-> ( ( ps /\ th ) \/ ( et /\ ( ps \/_ th ) ) ) )
9		df-cad 1546	. 2 |- (cadd ( ch , ta , ze ) <-> ( ( ch /\ ta ) \/ ( ze /\ ( ch \/_ ta ) ) ) )
10	7, 8, 9	3bitr4g 303	1 |- ( ph -> (cadd ( ps , th , et ) <-> cadd ( ch , ta , ze ) ) )

Syntax hints:   -> 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-xor 1465  df-cad 1546

This theorem is referenced by:  cadbi123i  1550  sadfval  15174  sadcp1  15177