Theorem anddi 840

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

Assertion

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

Proof of Theorem anddi

Step	Hyp	Ref	Expression
1		andir 838	. 2 |- (((ph \/ ps) /\ (ch \/ th)) <-> ((ph /\ (ch \/ th)) \/ (ps /\ (ch \/ th))))
2		andi 837	. . 3 |- ((ph /\ (ch \/ th)) <-> ((ph /\ ch) \/ (ph /\ th)))
3		andi 837	. . 3 |- ((ps /\ (ch \/ th)) <-> ((ps /\ ch) \/ (ps /\ th)))
4	2, 3	orbi12i 507	. 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:  funun  5147