Theorem orddi 605

Description: Double distributive law for disjunction.

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 596	. 2 |- (((ph /\ ps) \/ (ch /\ th)) <-> ((ph \/ (ch /\ th)) /\ (ps \/ (ch /\ th))))
2		ordi 595	. . 3 |- ((ph \/ (ch /\ th)) <-> ((ph \/ ch) /\ (ph \/ th)))
3		ordi 595	. . 3 |- ((ps \/ (ch /\ th)) <-> ((ps \/ ch) /\ (ps \/ th)))
4	2, 3	anbi12i 482	. 2 |- (((ph \/ (ch /\ th)) /\ (ps \/ (ch /\ th))) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))
5	1, 4	bitr 173	1 |- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  icounlem 6353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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