Theorem orddi 839

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

Assertion

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

Proof of Theorem orddi

Step	Hyp	Ref	Expression
1		ordir 835	. 2 |- (((ph /\ ps) \/ (ch /\ th)) <-> ((ph \/ (ch /\ th)) /\ (ps \/ (ch /\ th))))
2		ordi 834	. . 3 |- ((ph \/ (ch /\ th)) <-> ((ph \/ ch) /\ (ph \/ th)))
3		ordi 834	. . 3 |- ((ps \/ (ch /\ th)) <-> ((ps \/ ch) /\ (ps \/ th)))
4	2, 3	anbi12i 678	. 2 |- (((ph \/ (ch /\ th)) /\ (ps \/ (ch /\ th))) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))
5	1, 4	bitri 240	1 |- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))

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: (None)