Theorem 3bitrrd 271

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

Hypotheses

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

Assertion

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

Proof of Theorem 3bitrrd

Step	Hyp	Ref	Expression
1		3bitrd.3	. 2 |- (ph -> (th <-> ta))
2		3bitrd.1	. . 3 |- (ph -> (ps <-> ch))
3		3bitrd.2	. . 3 |- (ph -> (ch <-> th))
4	2, 3	bitr2d 245	. 2 |- (ph -> (th <-> ps))
5	1, 4	bitr3d 246	1 |- (ph -> (ta <-> ps))

This proof depends on syntax axioms:   -> wi 4   <-> wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by: (None)