ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl6rbb Unicode version

Theorem syl6rbb 190
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
StepHypRef Expression
1 syl6rbb.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
2 syl6rbb.2 . . 3  |-  ( ch  <->  th )
31, 2syl6bb 189 . 2  |-  ( ph  ->  ( ps  <->  th )
)
43bicomd 133 1  |-  ( ph  ->  ( th  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  syl6rbbr  192  bibif  624  pm5.61  718  oranabs  739  pm5.7dc  872  nbbndc  1301  resopab2  4682  xpcom  4891  f1od2  5883  ac6sfi  6382  elznn0  8316  rexuz3  9816
  Copyright terms: Public domain W3C validator