Theorem andir 457

Description: Distributive law for conjunction.

Assertion

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

Proof of Theorem andir

Step	Hyp	Ref	Expression
1		andi 456	. 2 |- ((ch /\ (ph \/ ps)) <-> ((ch /\ ph) \/ (ch /\ ps)))
2		ancom 333	. 2 |- (((ph \/ ps) /\ ch) <-> (ch /\ (ph \/ ps)))
3		ancom 333	. . 3 |- ((ph /\ ch) <-> (ch /\ ph))
4		ancom 333	. . 3 |- ((ps /\ ch) <-> (ch /\ ps))
5	3, 4	orbi12i 216	. 2 |- (((ph /\ ch) \/ (ps /\ ch)) <-> ((ch /\ ph) \/ (ch /\ ps)))
6	1, 2, 5	3bitr4 158	1 |- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))

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

This theorem is referenced by:  anddi 459  biass 511  caselem 561  iunxun 2035  xpundir 2462  nnmcan 3190

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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