Theorem 3bitrrd 215

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 189	. 2 |- ( ph -> ( th <-> ps ) )
5	1, 4	bitr3d 190	1 |- ( ph -> ( ta <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  srpospr  7867  divap0b  8727  divfl0  10403  cjreb  11048  eqg0el  13435  ghmeqker  13477  cnrest2  14556  2lgslem1a2  15412