Theorem cadcoma 1395

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

Assertion

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

Proof of Theorem cadcoma

Step	Hyp	Ref	Expression
1		ancom 437	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
2		xorcom 1307	. . . 4 |- ((ph \/_ ps) <-> (ps \/_ ph))
3	2	anbi2i 675	. . 3 |- ((ch /\ (ph \/_ ps)) <-> (ch /\ (ps \/_ ph)))
4	1, 3	orbi12i 507	. 2 |- (((ph /\ ps) \/ (ch /\ (ph \/_ ps))) <-> ((ps /\ ph) \/ (ch /\ (ps \/_ ph))))
5		df-cad 1381	. 2 |- (cadd(ph, ps, ch) <-> ((ph /\ ps) \/ (ch /\ (ph \/_ ps))))
6		df-cad 1381	. 2 |- (cadd(ps, ph, ch) <-> ((ps /\ ph) \/ (ch /\ (ps \/_ ph))))
7	4, 5, 6	3bitr4i 268	1 |- (cadd(ph, ps, ch) <-> cadd(ps, ph, ch))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358   \/_ 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-xor 1305  df-cad 1381

This theorem is referenced by:  cadrot  1397