Theorem 3bitr3d 217

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 189	. 2 |- ( ph -> ( th <-> ch ) )
4		3bitr3d.3	. 2 |- ( ph -> ( ch <-> ta ) )
5	3, 4	bitrd 187	1 |- ( ph -> ( th <-> ta ) )

Syntax hints:   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  csbcomg  3025  eloprabga  5858  ereldm  6472  mapen  6740  ordiso2  6920  subcan  8017  conjmulap  8489  ltrec  8641  divelunit  9785  fseq1m1p1  9875  fzm1  9880  fihashneq0  10541  hashfacen  10579  cvg1nlemcau  10756  lenegsq  10867  dvdsmod  11560  bezoutlemle  11696  rpexp  11831  qnumdenbi  11870  bdxmet  12670  txmetcnp  12687  cnmet  12699