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Theorem syl3an9b 1216
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 443 . . 3  |-  ( (
ph  /\  th )  ->  ( ps  <->  ta )
)
4 syl3an9b.3 . . 3  |-  ( et 
->  ( ta  <->  ze )
)
53, 4sylan9bb 443 . 2  |-  ( ( ( ph  /\  th )  /\  et )  -> 
( ps  <->  ze )
)
653impa 1110 1  |-  ( (
ph  /\  th  /\  et )  ->  ( ps  <->  ze )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896
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  df-3an 898
This theorem is referenced by:  eloprabg  5620
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