Theorem anddi 459

Description: Double distributive law for conjunction.

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 457	. 2 |- (((ph \/ ps) /\ (ch \/ th)) <-> ((ph /\ (ch \/ th)) \/ (ps /\ (ch \/ th))))
2		andi 456	. . 3 |- ((ph /\ (ch \/ th)) <-> ((ph /\ ch) \/ (ph /\ th)))
3		andi 456	. . 3 |- ((ps /\ (ch \/ th)) <-> ((ps /\ ch) \/ (ps /\ th)))
4	2, 3	orbi12i 216	. 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 is referenced by:  funun 2700

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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