Theorem 3bitr2rd 217

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
3bitr2rd	|- ( ph -> ( ta <-> ps ) )

Proof of Theorem 3bitr2rd

Step	Hyp	Ref	Expression
1		3bitr2d.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2		3bitr2d.2	. . 3 |- ( ph -> ( th <-> ch ) )
3	1, 2	bitr4d 191	. 2 |- ( ph -> ( ps <-> th ) )
4		3bitr2d.3	. 2 |- ( ph -> ( th <-> ta ) )
5	3, 4	bitr2d 189	1 |- ( ph -> ( ta <-> ps ) )

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:  pm4.55dc  951  anordc  969  fndmdif  5814  addsubeq4  8541  muleqadd  8999  nn0lt10b  9728  adddivflid  10729  frec2uzltd  10842  mul0inf  12009  summodnegmod  12591  lgsdilem  16158  lgsne0  16169  iooref1o  17095