Theorem andir 838

Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
andir	|- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))

Proof of Theorem andir

Step	Hyp	Ref	Expression
1		andi 837	. 2 |- ((ch /\ (ph \/ ps)) <-> ((ch /\ ph) \/ (ch /\ ps)))
2		ancom 437	. 2 |- (((ph \/ ps) /\ ch) <-> (ch /\ (ph \/ ps)))
3		ancom 437	. . 3 |- ((ph /\ ch) <-> (ch /\ ph))
4		ancom 437	. . 3 |- ((ps /\ ch) <-> (ch /\ ps))
5	3, 4	orbi12i 507	. 2 |- (((ph /\ ch) \/ (ps /\ ch)) <-> ((ch /\ ph) \/ (ch /\ ps)))
6	1, 2, 5	3bitr4i 268	1 |- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358

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

This theorem is referenced by:  anddi  840  cador  1391  rexun  3444  rabun2  3535  reuun2  3539  elimif  3692  xpundir  4834  coundi  5083  nchoicelem18  6307