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  699  pm5.61  795  oranabs  816  pm5.7dc  956  nbbndc  1413  resopab2  5003  xpcom  5226  f1od2  6311  map1  6889  ac6sfi  6977  elznn0  9369  rexuz3  11220  xrmaxiflemcom  11479  metrest  14896  sincosq3sgn  15218  sincosq4sgn  15219  lgsquadlem3  15474