ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl3an9b Unicode version

Theorem syl3an9b 1292
Description: Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
Hypotheses
Ref Expression
syl3an9b.1  |-  ( ph  ->  ( ps  <->  ch )
)
syl3an9b.2  |-  ( th 
->  ( ch  <->  ta )
)
syl3an9b.3  |-  ( et 
->  ( ta  <->  ze )
)
Assertion
Ref Expression
syl3an9b  |-  ( (
ph  /\  th  /\  et )  ->  ( ps  <->  ze )
)

Proof of Theorem syl3an9b
StepHypRef Expression
1 syl3an9b.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
2 syl3an9b.2 . . . 4  |-  ( th 
->  ( ch  <->  ta )
)
31, 2sylan9bb 458 . . 3  |-  ( (
ph  /\  th )  ->  ( ps  <->  ta )
)
4 syl3an9b.3 . . 3  |-  ( et 
->  ( ta  <->  ze )
)
53, 4sylan9bb 458 . 2  |-  ( ( ( ph  /\  th )  /\  et )  -> 
( ps  <->  ze )
)
653impa 1177 1  |-  ( (
ph  /\  th  /\  et )  ->  ( ps  <->  ze )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-3an 965
This theorem is referenced by:  eloprabg  5906
  Copyright terms: Public domain W3C validator