Theorem bitr2di 197

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 196	. 2 |- ( ph -> ( ps <-> th ) )
4	3	bicomd 141	1 |- ( ph -> ( th <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bitr4id  199  bibif  700  pm5.61  796  oranabs  817  pm5.7dc  957  nbbndc  1414  resopab2  5025  xpcom  5248  f1od2  6344  map1  6928  ac6sfi  7021  elznn0  9422  rexuz3  11416  xrmaxiflemcom  11675  metrest  15093  sincosq3sgn  15415  sincosq4sgn  15416  lgsquadlem3  15671  pw1map  16134