Theorem hadbi123d 1382

Description: Equality theorem for half adder. (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
hadbi123d	|- (ph -> (hadd(ps, th, et) <-> hadd(ch, ta, ze)))

Proof of Theorem hadbi123d

Step	Hyp	Ref	Expression
1		hadbid.1	. . . 4 |- (ph -> (ps <-> ch))
2		hadbid.2	. . . 4 |- (ph -> (th <-> ta))
3	1, 2	xorbi12d 1315	. . 3 |- (ph -> ((ps \/_ th) <-> (ch \/_ ta)))
4		hadbid.3	. . 3 |- (ph -> (et <-> ze))
5	3, 4	xorbi12d 1315	. 2 |- (ph -> (((ps \/_ th) \/_ et) <-> ((ch \/_ ta) \/_ ze)))
6		df-had 1380	. 2 |- (hadd(ps, th, et) <-> ((ps \/_ th) \/_ et))
7		df-had 1380	. 2 |- (hadd(ch, ta, ze) <-> ((ch \/_ ta) \/_ ze))
8	5, 6, 7	3bitr4g 279	1 |- (ph -> (hadd(ps, th, et) <-> hadd(ch, ta, ze)))

Syntax hints:   -> wi 4   <-> wb 176   \/_ wxo 1304  haddwhad 1378

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-xor 1305  df-had 1380

This theorem is referenced by:  hadbi123i  1384