Theorem 3bitr3d 207

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)

Hypotheses

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

Assertion

Ref	Expression
3bitr3d	|- ( ph -> ( th <-> ta ) )

Proof of Theorem 3bitr3d

Step	Hyp	Ref	Expression
1		3bitr3d.2	. . 3 |- ( ph -> ( ps <-> th ) )
2		3bitr3d.1	. . 3 |- ( ph -> ( ps <-> ch ) )
3	1, 2	bitr3d 179	. 2 |- ( ph -> ( th <-> ch ) )
4		3bitr3d.3	. 2 |- ( ph -> ( ch <-> ta ) )
5	3, 4	bitrd 177	1 |- ( ph -> ( th <-> ta ) )

Syntax hints:   -> wi 4   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  csbcomg  2873  eloprabga  5591  ereldm  6149  ordiso2  6357  subcan  7266  conjmulap  7705  ltrec  7849  divelunit  8870  fseq1m1p1  8957  fzm1  8962  cvg1nlemcau  9583  lenegsq  9691