Theorem 3bitr3rd 219

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

Hypotheses

Ref	Expression
3bitr3d.1	|- ( ph -> ( ps <-> ch ) )
3bitr3d.2	|- ( ph -> ( ps <-> th ) )
3bitr3d.3	|- ( ph -> ( ch <-> ta ) )

Assertion

Ref	Expression
3bitr3rd	|- ( ph -> ( ta <-> th ) )

Proof of Theorem 3bitr3rd

Step	Hyp	Ref	Expression
1		3bitr3d.3	. 2 |- ( ph -> ( ch <-> ta ) )
2		3bitr3d.1	. . 3 |- ( ph -> ( ps <-> ch ) )
3		3bitr3d.2	. . 3 |- ( ph -> ( ps <-> th ) )
4	2, 3	bitr3d 190	. 2 |- ( ph -> ( ch <-> th ) )
5	1, 4	bitr3d 190	1 |- ( ph -> ( ta <-> th ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  funconstss  5827  eqneg  9065  minclpr  12020  ballotfilemrv  13314  evenennn  13335  nmzsubg  14064  znidomb  15044  rpcxple2  16076  rpcxplt2  16077  wilthlem1  16154  lgslem1  16241