Theorem ordi 729

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 102	. . . 4 |- ( ( ps /\ ch ) -> ps )
2	1	orim2i 678	. . 3 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ph \/ ps ) )
3		simpr 103	. . . 4 |- ( ( ps /\ ch ) -> ch )
4	3	orim2i 678	. . 3 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ph \/ ch ) )
5	2, 4	jca 290	. 2 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )
6		orc 633	. . . 4 |- ( ph -> ( ph \/ ( ps /\ ch ) ) )
7	6	adantl 262	. . 3 |- ( ( ( ph \/ ps ) /\ ph ) -> ( ph \/ ( ps /\ ch ) ) )
8	6	adantr 261	. . . 4 |- ( ( ph /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
9		olc 632	. . . 4 |- ( ( ps /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
10	8, 9	jaoian 709	. . 3 |- ( ( ( ph \/ ps ) /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
11	7, 10	jaodan 710	. 2 |- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) -> ( ph \/ ( ps /\ ch ) ) )
12	5, 11	impbii 117	1 |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )

Syntax hints:   /\ wa 97   <-> wb 98   \/ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ordir  730  orddi  733  pm5.63dc  853  pm4.43  856  orbididc  860  undi  3185  undif4  3284  elnn1uz2  8544