Theorem cadbi123i 1550

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

Hypotheses

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

Assertion

Ref	Expression
cadbi123i	|- (cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) )

Proof of Theorem cadbi123i

Step	Hyp	Ref	Expression
1		cadbii.1	. . . 4 |- ( ph <-> ps )
2	1	a1i 11	. . 3 |- ( T. -> ( ph <-> ps ) )
3		cadbii.2	. . . 4 |- ( ch <-> th )
4	3	a1i 11	. . 3 |- ( T. -> ( ch <-> th ) )
5		cadbii.3	. . . 4 |- ( ta <-> et )
6	5	a1i 11	. . 3 |- ( T. -> ( ta <-> et ) )
7	2, 4, 6	cadbi123d 1549	. 2 |- ( T. -> (cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) ) )
8	7	trud 1493	1 |- (cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) )

Syntax hints:   <-> wb 196   T. wtru 1484  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-tru 1486  df-cad 1546

This theorem is referenced by: (None)