Theorem ordir 812

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 811	. 2 |- ( ( ch \/ ( ph /\ ps ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
2		orcom 723	. 2 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ch \/ ( ph /\ ps ) ) )
3		orcom 723	. . 3 |- ( ( ph \/ ch ) <-> ( ch \/ ph ) )
4		orcom 723	. . 3 |- ( ( ps \/ ch ) <-> ( ch \/ ps ) )
5	3, 4	anbi12i 457	. 2 |- ( ( ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
6	1, 2, 5	3bitr4i 211	1 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  orddi  815  pm5.62dc  940  dn1dc  955  suc11g  4541  bj-peano4  13990