Theorem cadbi123d 1383

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

Hypotheses

Ref	Expression
hadbid.1	|- (ph -> (ps <-> ch))
hadbid.2	|- (ph -> (th <-> ta))
hadbid.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		hadbid.1	. . . 4 |- (ph -> (ps <-> ch))
2		hadbid.2	. . . 4 |- (ph -> (th <-> ta))
3	1, 2	anbi12d 691	. . 3 |- (ph -> ((ps /\ th) <-> (ch /\ ta)))
4		hadbid.3	. . . 4 |- (ph -> (et <-> ze))
5	1, 2	xorbi12d 1315	. . . 4 |- (ph -> ((ps \/_ th) <-> (ch \/_ ta)))
6	4, 5	anbi12d 691	. . 3 |- (ph -> ((et /\ (ps \/_ th)) <-> (ze /\ (ch \/_ ta))))
7	3, 6	orbi12d 690	. 2 |- (ph -> (((ps /\ th) \/ (et /\ (ps \/_ th))) <-> ((ch /\ ta) \/ (ze /\ (ch \/_ ta)))))
8		df-cad 1381	. 2 |- (cadd(ps, th, et) <-> ((ps /\ th) \/ (et /\ (ps \/_ th))))
9		df-cad 1381	. 2 |- (cadd(ch, ta, ze) <-> ((ch /\ ta) \/ (ze /\ (ch \/_ ta))))
10	7, 8, 9	3bitr4g 279	1 |- (ph -> (cadd(ps, th, et) <-> cadd(ch, ta, ze)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   \/_ wxo 1304  caddwcad 1379

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 177  df-or 359  df-an 360  df-xor 1305  df-cad 1381

This theorem is referenced by:  cadbi123i  1385