Theorem orddi 458

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 453	. 2 |- (((ph /\ ps) \/ (ch /\ th)) <-> ((ph \/ (ch /\ th)) /\ (ps \/ (ch /\ th))))
2		ordi 452	. . 3 |- ((ph \/ (ch /\ th)) <-> ((ph \/ ch) /\ (ph \/ th)))
3		ordi 452	. . 3 |- ((ps \/ (ch /\ th)) <-> ((ps \/ ch) /\ (ps \/ th)))
4	2, 3	anbi12i 369	. 2 |- (((ph \/ (ch /\ th)) /\ (ps \/ (ch /\ th))) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))
5	1, 4	bitr 151	1 |- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))

Syntax hints:   <-> wb 127   \/ wo 195   /\ wa 196

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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