Theorem or3di 29307

Description: Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)

Assertion

Ref	Expression
or3di	|- ( ( ph \/ ( ps /\ ch /\ ta ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ph \/ ta ) ) )

Proof of Theorem or3di

Step	Hyp	Ref	Expression
1		df-3an 1039	. . . 4 |- ( ( ps /\ ch /\ ta ) <-> ( ( ps /\ ch ) /\ ta ) )
2	1	orbi2i 541	. . 3 |- ( ( ph \/ ( ps /\ ch /\ ta ) ) <-> ( ph \/ ( ( ps /\ ch ) /\ ta ) ) )
3		ordi 908	. . 3 |- ( ( ph \/ ( ( ps /\ ch ) /\ ta ) ) <-> ( ( ph \/ ( ps /\ ch ) ) /\ ( ph \/ ta ) ) )
4		ordi 908	. . . 4 |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )
5	4	anbi1i 731	. . 3 |- ( ( ( ph \/ ( ps /\ ch ) ) /\ ( ph \/ ta ) ) <-> ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ph \/ ta ) ) )
6	2, 3, 5	3bitri 286	. 2 |- ( ( ph \/ ( ps /\ ch /\ ta ) ) <-> ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ph \/ ta ) ) )
7		df-3an 1039	. 2 |- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ph \/ ta ) ) <-> ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ph \/ ta ) ) )
8	6, 7	bitr4i 267	1 |- ( ( ph \/ ( ps /\ ch /\ ta ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ph \/ ta ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039

This theorem is referenced by:  or3dir  29308