Theorem ordi 762

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

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

Proof of Theorem ordi

Step	Hyp	Ref	Expression
1		simpl 107	. . . 4 |- ( ( ps /\ ch ) -> ps )
2	1	orim2i 710	. . 3 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ph \/ ps ) )
3		simpr 108	. . . 4 |- ( ( ps /\ ch ) -> ch )
4	3	orim2i 710	. . 3 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ph \/ ch ) )
5	2, 4	jca 300	. 2 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )
6		orc 665	. . . 4 |- ( ph -> ( ph \/ ( ps /\ ch ) ) )
7	6	adantl 271	. . 3 |- ( ( ( ph \/ ps ) /\ ph ) -> ( ph \/ ( ps /\ ch ) ) )
8	6	adantr 270	. . . 4 |- ( ( ph /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
9		olc 664	. . . 4 |- ( ( ps /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
10	8, 9	jaoian 741	. . 3 |- ( ( ( ph \/ ps ) /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
11	7, 10	jaodan 743	. 2 |- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) -> ( ph \/ ( ps /\ ch ) ) )
12	5, 11	impbii 124	1 |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ 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:  ordir  763  orddi  766  pm5.63dc  887  pm4.43  890  orbididc  894  undi  3212  undif4  3306  elnn1uz2  8694