Theorem 3an4anass 1291

Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)

Assertion

Ref	Expression
3an4anass	|- ( ( ( ph /\ ps /\ ch ) /\ th ) <-> ( ( ph /\ ps ) /\ ( ch /\ th ) ) )

Proof of Theorem 3an4anass

Step	Hyp	Ref	Expression
1		df-3an 1039	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2	1	anbi1i 731	. 2 |- ( ( ( ph /\ ps /\ ch ) /\ th ) <-> ( ( ( ph /\ ps ) /\ ch ) /\ th ) )
3		anass 681	. 2 |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) <-> ( ( ph /\ ps ) /\ ( ch /\ th ) ) )
4	2, 3	bitri 264	1 |- ( ( ( ph /\ ps /\ ch ) /\ th ) <-> ( ( ph /\ ps ) /\ ( ch /\ 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:  oeeui  7682  isclwlkupgr  26674  clwlkclwwlk  26903  bnj557  30971