Theorem ordi 805

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 108	. . . 4 |- ( ( ps /\ ch ) -> ps )
2	1	orim2i 750	. . 3 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ph \/ ps ) )
3		simpr 109	. . . 4 |- ( ( ps /\ ch ) -> ch )
4	3	orim2i 750	. . 3 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ph \/ ch ) )
5	2, 4	jca 304	. 2 |- ( ( ph \/ ( ps /\ ch ) ) -> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )
6		orc 701	. . . 4 |- ( ph -> ( ph \/ ( ps /\ ch ) ) )
7	6	adantl 275	. . 3 |- ( ( ( ph \/ ps ) /\ ph ) -> ( ph \/ ( ps /\ ch ) ) )
8	6	adantr 274	. . . 4 |- ( ( ph /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
9		olc 700	. . . 4 |- ( ( ps /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
10	8, 9	jaoian 784	. . 3 |- ( ( ( ph \/ ps ) /\ ch ) -> ( ph \/ ( ps /\ ch ) ) )
11	7, 10	jaodan 786	. 2 |- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) -> ( ph \/ ( ps /\ ch ) ) )
12	5, 11	impbii 125	1 |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 697

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 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ordir  806  orddi  809  pm5.63dc  930  pm4.43  933  orbididc  937  undi  3324  undif4  3425  elnn1uz2  9401