Theorem cador 1391

Description: Write the adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
cador	|- (cadd(ph, ps, ch) <-> ((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)))

Proof of Theorem cador

Step	Hyp	Ref	Expression
1		df-cad 1381	. 2 |- (cadd(ph, ps, ch) <-> ((ph /\ ps) \/ (ch /\ (ph \/_ ps))))
2		xor2 1310	. . . . . . 7 |- ((ph \/_ ps) <-> ((ph \/ ps) /\ -. (ph /\ ps)))
3	2	rbaib 873	. . . . . 6 |- (-. (ph /\ ps) -> ((ph \/_ ps) <-> (ph \/ ps)))
4	3	anbi1d 685	. . . . 5 |- (-. (ph /\ ps) -> (((ph \/_ ps) /\ ch) <-> ((ph \/ ps) /\ ch)))
5		ancom 437	. . . . 5 |- (((ph \/_ ps) /\ ch) <-> (ch /\ (ph \/_ ps)))
6		andir 838	. . . . 5 |- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))
7	4, 5, 6	3bitr3g 278	. . . 4 |- (-. (ph /\ ps) -> ((ch /\ (ph \/_ ps)) <-> ((ph /\ ch) \/ (ps /\ ch))))
8	7	pm5.74i 236	. . 3 |- ((-. (ph /\ ps) -> (ch /\ (ph \/_ ps))) <-> (-. (ph /\ ps) -> ((ph /\ ch) \/ (ps /\ ch))))
9		df-or 359	. . 3 |- (((ph /\ ps) \/ (ch /\ (ph \/_ ps))) <-> (-. (ph /\ ps) -> (ch /\ (ph \/_ ps))))
10		3orass 937	. . . 4 |- (((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)) <-> ((ph /\ ps) \/ ((ph /\ ch) \/ (ps /\ ch))))
11		df-or 359	. . . 4 |- (((ph /\ ps) \/ ((ph /\ ch) \/ (ps /\ ch))) <-> (-. (ph /\ ps) -> ((ph /\ ch) \/ (ps /\ ch))))
12	10, 11	bitri 240	. . 3 |- (((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)) <-> (-. (ph /\ ps) -> ((ph /\ ch) \/ (ps /\ ch))))
13	8, 9, 12	3bitr4i 268	. 2 |- (((ph /\ ps) \/ (ch /\ (ph \/_ ps))) <-> ((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)))
14	1, 13	bitri 240	1 |- (cadd(ph, ps, ch) <-> ((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   \/ w3o 933   \/_ 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-3or 935  df-xor 1305  df-cad 1381

This theorem is referenced by:  cadan  1392  cadnot  1394  cadcomb  1396