Theorem syl6rbb 186

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

Hypotheses

Ref	Expression
syl6rbb.1	|- ( ph -> ( ps <-> ch ) )
syl6rbb.2	|- ( ch <-> th )

Assertion

Ref	Expression
syl6rbb	|- ( ph -> ( th <-> ps ) )

Proof of Theorem syl6rbb

Step	Hyp	Ref	Expression
1		syl6rbb.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2		syl6rbb.2	. . 3 |- ( ch <-> th )
3	1, 2	syl6bb 185	. 2 |- ( ph -> ( ps <-> th ) )
4	3	bicomd 129	1 |- ( ph -> ( th <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  syl6rbbr  188  bibif  614  pm5.61  708  oranabs  728  pm5.7dc  861  nbbndc  1285  resopab2  4655  xpcom  4864  ac6sfi  6352  elznn0  8260  rexuz3  9588