Theorem andir 604

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 603	. 2 |- ((ch /\ (ph \/ ps)) <-> ((ch /\ ph) \/ (ch /\ ps)))
2		ancom 435	. 2 |- (((ph \/ ps) /\ ch) <-> (ch /\ (ph \/ ps)))
3		ancom 435	. . 3 |- ((ph /\ ch) <-> (ch /\ ph))
4		ancom 435	. . 3 |- ((ps /\ ch) <-> (ch /\ ps))
5	3, 4	orbi12i 257	. 2 |- (((ph /\ ch) \/ (ps /\ ch)) <-> ((ch /\ ph) \/ (ch /\ ps)))
6	1, 2, 5	3bitr4 183	1 |- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))

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

This theorem is referenced by:  anddi 606  reuun2 2274  elimif 2370  iunxun 2609  xpundir 3221  fopabap 3832  oarec 4186  snunioolem 6355  pilem1 8609

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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