Metamath Proof Explorer


Theorem 3bitr2d

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006)

Ref Expression
Hypotheses 3bitr2d.1
|- ( ph -> ( ps <-> ch ) )
3bitr2d.2
|- ( ph -> ( th <-> ch ) )
3bitr2d.3
|- ( ph -> ( th <-> ta ) )
Assertion 3bitr2d
|- ( ph -> ( ps <-> ta ) )

Proof

Step Hyp Ref Expression
1 3bitr2d.1
 |-  ( ph -> ( ps <-> ch ) )
2 3bitr2d.2
 |-  ( ph -> ( th <-> ch ) )
3 3bitr2d.3
 |-  ( ph -> ( th <-> ta ) )
4 1 2 bitr4d
 |-  ( ph -> ( ps <-> th ) )
5 4 3 bitrd
 |-  ( ph -> ( ps <-> ta ) )