Theorem cadbi123i 1385

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

Hypotheses

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

Assertion

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

Proof of Theorem cadbi123i

Step	Hyp	Ref	Expression
1		hadbii.1	. . . 4 |- (ph <-> ps)
2	1	a1i 10	. . 3 |- ( T. -> (ph <-> ps))
3		hadbii.2	. . . 4 |- (ch <-> th)
4	3	a1i 10	. . 3 |- ( T. -> (ch <-> th))
5		hadbii.3	. . . 4 |- (ta <-> et)
6	5	a1i 10	. . 3 |- ( T. -> (ta <-> et))
7	2, 4, 6	cadbi123d 1383	. 2 |- ( T. -> (cadd(ph, ch, ta) <-> cadd(ps, th, et)))
8	7	trud 1323	1 |- (cadd(ph, ch, ta) <-> cadd(ps, th, et))

Syntax hints:   <-> wb 176   T. wtru 1316  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-tru 1319  df-cad 1381

This theorem is referenced by: (None)