Theorem bitr2di 196

Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
bitr2di.1	|- ( ph -> ( ps <-> ch ) )
bitr2di.2	|- ( ch <-> th )

Assertion

Ref	Expression
bitr2di	|- ( ph -> ( th <-> ps ) )

Proof of Theorem bitr2di

Step	Hyp	Ref	Expression
1		bitr2di.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2		bitr2di.2	. . 3 |- ( ch <-> th )
3	1, 2	bitrdi 195	. 2 |- ( ph -> ( ps <-> th ) )
4	3	bicomd 140	1 |- ( ph -> ( th <-> ps ) )

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:  bitr4id  198  bibif  693  pm5.61  789  oranabs  810  pm5.7dc  949  nbbndc  1389  resopab2  4938  xpcom  5157  f1od2  6214  map1  6790  ac6sfi  6876  elznn0  9227  rexuz3  10954  xrmaxiflemcom  11212  metrest  13300  sincosq3sgn  13543  sincosq4sgn  13544