Theorem an3andi 1445

Description: Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)

Assertion

Ref	Expression
an3andi	|- ( ( ph /\ ( ps /\ ch /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) ) )

Proof of Theorem an3andi

Step	Hyp	Ref	Expression
1		df-3an 1039	. . . 4 |- ( ( ps /\ ch /\ th ) <-> ( ( ps /\ ch ) /\ th ) )
2	1	anbi2i 730	. . 3 |- ( ( ph /\ ( ps /\ ch /\ th ) ) <-> ( ph /\ ( ( ps /\ ch ) /\ th ) ) )
3		anandi 871	. . 3 |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) <-> ( ( ph /\ ( ps /\ ch ) ) /\ ( ph /\ th ) ) )
4		anandi 871	. . . 4 |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )
5	4	anbi1i 731	. . 3 |- ( ( ( ph /\ ( ps /\ ch ) ) /\ ( ph /\ th ) ) <-> ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) /\ ( ph /\ th ) ) )
6	2, 3, 5	3bitri 286	. 2 |- ( ( ph /\ ( ps /\ ch /\ th ) ) <-> ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) /\ ( ph /\ th ) ) )
7		df-3an 1039	. 2 |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) ) <-> ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) /\ ( ph /\ th ) ) )
8	6, 7	bitr4i 267	1 |- ( ( ph /\ ( ps /\ ch /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) ) )

Syntax hints:   <-> wb 196   /\ 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-an 386  df-3an 1039

This theorem is referenced by:  raltpd  4315