Theorem or3dir 29308

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

Assertion

Ref	Expression
or3dir	|- ( ( ( ph /\ ps /\ ch ) \/ ta ) <-> ( ( ph \/ ta ) /\ ( ps \/ ta ) /\ ( ch \/ ta ) ) )

Proof of Theorem or3dir

Step	Hyp	Ref	Expression
1		or3di 29307	. 2 |- ( ( ta \/ ( ph /\ ps /\ ch ) ) <-> ( ( ta \/ ph ) /\ ( ta \/ ps ) /\ ( ta \/ ch ) ) )
2		orcom 402	. 2 |- ( ( ta \/ ( ph /\ ps /\ ch ) ) <-> ( ( ph /\ ps /\ ch ) \/ ta ) )
3		orcom 402	. . 3 |- ( ( ta \/ ph ) <-> ( ph \/ ta ) )
4		orcom 402	. . 3 |- ( ( ta \/ ps ) <-> ( ps \/ ta ) )
5		orcom 402	. . 3 |- ( ( ta \/ ch ) <-> ( ch \/ ta ) )
6	3, 4, 5	3anbi123i 1251	. 2 |- ( ( ( ta \/ ph ) /\ ( ta \/ ps ) /\ ( ta \/ ch ) ) <-> ( ( ph \/ ta ) /\ ( ps \/ ta ) /\ ( ch \/ ta ) ) )
7	1, 2, 6	3bitr3i 290	1 |- ( ( ( ph /\ ps /\ ch ) \/ ta ) <-> ( ( ph \/ ta ) /\ ( ps \/ ta ) /\ ( ch \/ ta ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ 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: (None)