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  909  dn1dc  966  csbabg  3186  uniiunlem  3313  inimasn  5145  cnvpom  5270  fnresdisj  5432  f1oiso  5949  reldm  6330  mptelixpg  6879  1idprl  7773  1idpru  7774  nndiv  9147  fzn  10234  fz1sbc  10288  grpid  13567  znleval  14611  metrest  15174