Theorem ordir 835

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

Assertion

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

Proof of Theorem ordir

Step	Hyp	Ref	Expression
1		ordi 834	. 2 |- ((ch \/ (ph /\ ps)) <-> ((ch \/ ph) /\ (ch \/ ps)))
2		orcom 376	. 2 |- (((ph /\ ps) \/ ch) <-> (ch \/ (ph /\ ps)))
3		orcom 376	. . 3 |- ((ph \/ ch) <-> (ch \/ ph))
4		orcom 376	. . 3 |- ((ps \/ ch) <-> (ch \/ ps))
5	3, 4	anbi12i 678	. 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:  orddi  839  pm5.62  889  dn1  932  cadan  1392  nchoicelem9  6298