Theorem 3bitr2d 272

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

Hypotheses

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

Assertion

Ref	Expression
3bitr2d	|- (ph -> (ps <-> ta))

Proof of Theorem 3bitr2d

Step	Hyp	Ref	Expression
1		3bitr2d.1	. . 3 |- (ph -> (ps <-> ch))
2		3bitr2d.2	. . 3 |- (ph -> (th <-> ch))
3	1, 2	bitr4d 247	. 2 |- (ph -> (ps <-> th))
4		3bitr2d.3	. 2 |- (ph -> (th <-> ta))
5	3, 4	bitrd 244	1 |- (ph -> (ps <-> ta))

Syntax hints:   -> wi 4   <-> wb 176

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 177

This theorem is referenced by:  ceqsralt  2882