Theorem bitr2id 193

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

Hypotheses

Ref	Expression
bitr2id.1	|- ( ph <-> ps )
bitr2id.2	|- ( ch -> ( ps <-> th ) )

Assertion

Ref	Expression
bitr2id	|- ( ch -> ( th <-> ph ) )

Proof of Theorem bitr2id

Step	Hyp	Ref	Expression
1		bitr2id.1	. . 3 |- ( ph <-> ps )
2		bitr2id.2	. . 3 |- ( ch -> ( ps <-> th ) )
3	1, 2	bitrid 192	. 2 |- ( ch -> ( ph <-> th ) )
4	3	bicomd 141	1 |- ( ch -> ( th <-> ph ) )

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:  bitr3di  195  pm5.17dc  912  dn1dc  969  csbabg  3190  uniiunlem  3318  inimasn  5161  cnvpom  5286  fnresdisj  5449  f1oiso  5977  reldm  6358  mptelixpg  6946  1idprl  7853  1idpru  7854  nndiv  9226  fzn  10322  fz1sbc  10376  grpid  13685  znleval  14732  metrest  15300  loopclwwlkn1b  16343  clwwlknun  16365