Metamath Proof Explorer


Theorem 3bitr4d

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995)

Ref Expression
Hypotheses 3bitr4d.1
|- ( ph -> ( ps <-> ch ) )
3bitr4d.2
|- ( ph -> ( th <-> ps ) )
3bitr4d.3
|- ( ph -> ( ta <-> ch ) )
Assertion 3bitr4d
|- ( ph -> ( th <-> ta ) )

Proof

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