Theorem hadbi123d 1534

Description: Equality theorem for the adder sum. (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 1478	. . 3 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )
4		hadbid.3	. . 3 |- ( ph -> ( et <-> ze ) )
5	3, 4	xorbi12d 1478	. 2 |- ( ph -> ( ( ( ps \/_ th ) \/_ et ) <-> ( ( ch \/_ ta ) \/_ ze ) ) )
6		df-had 1533	. 2 |- (hadd ( ps , th , et ) <-> ( ( ps \/_ th ) \/_ et ) )
7		df-had 1533	. 2 |- (hadd ( ch , ta , ze ) <-> ( ( ch \/_ ta ) \/_ ze ) )
8	5, 6, 7	3bitr4g 303	1 |- ( ph -> (hadd ( ps , th , et ) <-> hadd ( ch , ta , ze ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/_ wxo 1464  haddwhad 1532

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-xor 1465  df-had 1533

This theorem is referenced by:  hadbi123i  1535  sadfval  15174  sadval  15178