Metamath Proof Explorer

Theorem bnj345

Description: /\ -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Andrew Salmon, 14-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion bnj345 
|- ( ( ph /\ ps /\ ch /\ th ) <-> ( th /\ ph /\ ps /\ ch ) )

Proof

Step Hyp Ref Expression
1 bnj334 
 |-  ( ( ph /\ ps /\ ch /\ th ) <-> ( ch /\ ph /\ ps /\ th ) )
2 bnj250 
 |-  ( ( ch /\ ph /\ ps /\ th ) <-> ( ch /\ ( ( ph /\ ps ) /\ th ) ) )
3 3anass 
 |-  ( ( ch /\ ( ph /\ ps ) /\ th ) <-> ( ch /\ ( ( ph /\ ps ) /\ th ) ) )
4 2 3 bitr4i 
 |-  ( ( ch /\ ph /\ ps /\ th ) <-> ( ch /\ ( ph /\ ps ) /\ th ) )
5 3anrev 
 |-  ( ( ch /\ ( ph /\ ps ) /\ th ) <-> ( th /\ ( ph /\ ps ) /\ ch ) )
6 bnj250 
 |-  ( ( th /\ ph /\ ps /\ ch ) <-> ( th /\ ( ( ph /\ ps ) /\ ch ) ) )
7 3anass 
 |-  ( ( th /\ ( ph /\ ps ) /\ ch ) <-> ( th /\ ( ( ph /\ ps ) /\ ch ) ) )
8 6 7 bitr4i 
 |-  ( ( th /\ ph /\ ps /\ ch ) <-> ( th /\ ( ph /\ ps ) /\ ch ) )
9 5 8 bitr4i 
 |-  ( ( ch /\ ( ph /\ ps ) /\ th ) <-> ( th /\ ph /\ ps /\ ch ) )
10 1 4 9 3bitri 
 |-  ( ( ph /\ ps /\ ch /\ th ) <-> ( th /\ ph /\ ps /\ ch ) )