Theorem ordir 829

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 828	. 2 |- ( ( ch \/ ( ph /\ ps ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
2		orcom 740	. 2 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ch \/ ( ph /\ ps ) ) )
3		orcom 740	. . 3 |- ( ( ph \/ ch ) <-> ( ch \/ ph ) )
4		orcom 740	. . 3 |- ( ( ps \/ ch ) <-> ( ch \/ ps ) )
5	3, 4	anbi12i 464	. 2 |- ( ( ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ch \/ ph ) /\ ( ch \/ ps ) ) )
6	1, 2, 5	3bitr4i 212	1 |- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   \/ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  orddi  832  dcand  945  pm5.62dc  958  dn1dc  973  suc11g  4704  bj-peano4  16981