Theorem anan 33999

Description: Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)

Assertion

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

Proof of Theorem anan

Step	Hyp	Ref	Expression
1		an4 865	. 2 |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ( ( ph /\ th ) /\ ta ) ) <-> ( ( ( ph /\ ps ) /\ ( ph /\ th ) ) /\ ( ch /\ ta ) ) )
2		anandi 871	. . . 4 |- ( ( ph /\ ( ps /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ th ) ) )
3		ancom 466	. . . 4 |- ( ( ph /\ ( ps /\ th ) ) <-> ( ( ps /\ th ) /\ ph ) )
4	2, 3	bitr3i 266	. . 3 |- ( ( ( ph /\ ps ) /\ ( ph /\ th ) ) <-> ( ( ps /\ th ) /\ ph ) )
5	4	anbi1i 731	. 2 |- ( ( ( ( ph /\ ps ) /\ ( ph /\ th ) ) /\ ( ch /\ ta ) ) <-> ( ( ( ps /\ th ) /\ ph ) /\ ( ch /\ ta ) ) )
6		anass 681	. 2 |- ( ( ( ( ps /\ th ) /\ ph ) /\ ( ch /\ ta ) ) <-> ( ( ps /\ th ) /\ ( ph /\ ( ch /\ ta ) ) ) )
7	1, 5, 6	3bitri 286	1 |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ( ( ph /\ th ) /\ ta ) ) <-> ( ( ps /\ th ) /\ ( ph /\ ( ch /\ ta ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384

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

This theorem is referenced by: (None)