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  706  pm5.61  802  oranabs  823  pm5.7dc  963  nbbndc  1439  resopab2  5066  xpcom  5290  f1od2  6409  map1  7030  ac6sfi  7130  elznn0  9538  rexuz3  11613  xrmaxiflemcom  11872  metrest  15300  sincosq3sgn  15622  sincosq4sgn  15623  lgsquadlem3  15881  pw1map  16700