Theorem ordir 763

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 762	. 2 |- ( ( ch \/ ( ph /\ ps ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
2		orcom 679	. 2 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ch \/ ( ph /\ ps ) ) )
3		orcom 679	. . 3 |- ( ( ph \/ ch ) <-> ( ch \/ ph ) )
4		orcom 679	. . 3 |- ( ( ps \/ ch ) <-> ( ch \/ ps ) )
5	3, 4	anbi12i 447	. 2 |- ( ( ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
6	1, 2, 5	3bitr4i 210	1 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )

Syntax hints:   /\ wa 102   <-> wb 103   \/ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  orddi  766  pm5.62dc  886  dn1dc  901  suc11g  4300  bj-peano4  10750