Theorem ordir 909

Description: Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)

Assertion

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

Proof of Theorem ordir

Step	Hyp	Ref	Expression
1		ordi 908	. 2 |- ( ( ch \/ ( ph /\ ps ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
2		orcom 402	. 2 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ch \/ ( ph /\ ps ) ) )
3		orcom 402	. . 3 |- ( ( ph \/ ch ) <-> ( ch \/ ph ) )
4		orcom 402	. . 3 |- ( ( ps \/ ch ) <-> ( ch \/ ps ) )
5	3, 4	anbi12i 733	. 2 |- ( ( ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
6	1, 2, 5	3bitr4i 292	1 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  orddi  913  pm5.62  958  dn1  1008  cadan  1548  elnn0z  11390  ifpim123g  37845  rp-fakeanorass  37858