Theorem cadcomb 1396

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

Assertion

Ref	Expression
cadcomb	|- (cadd(ph, ps, ch) <-> cadd(ph, ch, ps))

Proof of Theorem cadcomb

Step	Hyp	Ref	Expression
1		3orcoma 942	. . 3 |- (((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)) <-> ((ph /\ ch) \/ (ph /\ ps) \/ (ps /\ ch)))
2		biid 227	. . . 4 |- ((ph /\ ch) <-> (ph /\ ch))
3		biid 227	. . . 4 |- ((ph /\ ps) <-> (ph /\ ps))
4		ancom 437	. . . 4 |- ((ps /\ ch) <-> (ch /\ ps))
5	2, 3, 4	3orbi123i 1141	. . 3 |- (((ph /\ ch) \/ (ph /\ ps) \/ (ps /\ ch)) <-> ((ph /\ ch) \/ (ph /\ ps) \/ (ch /\ ps)))
6	1, 5	bitri 240	. 2 |- (((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)) <-> ((ph /\ ch) \/ (ph /\ ps) \/ (ch /\ ps)))
7		cador 1391	. 2 |- (cadd(ph, ps, ch) <-> ((ph /\ ps) \/ (ph /\ ch) \/ (ps /\ ch)))
8		cador 1391	. 2 |- (cadd(ph, ch, ps) <-> ((ph /\ ch) \/ (ph /\ ps) \/ (ch /\ ps)))
9	6, 7, 8	3bitr4i 268	1 |- (cadd(ph, ps, ch) <-> cadd(ph, ch, ps))

Syntax hints:   <-> wb 176   /\ wa 358   \/ w3o 933  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:  cadrot  1397