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Theorem syl3anb 1213
Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
Hypotheses
Ref Expression
syl3anb.1  |-  ( ph  <->  ps )
syl3anb.2  |-  ( ch  <->  th )
syl3anb.3  |-  ( ta  <->  et )
syl3anb.4  |-  ( ( ps  /\  th  /\  et )  ->  ze )
Assertion
Ref Expression
syl3anb  |-  ( (
ph  /\  ch  /\  ta )  ->  ze )

Proof of Theorem syl3anb
StepHypRef Expression
1 syl3anb.1 . . 3  |-  ( ph  <->  ps )
2 syl3anb.2 . . 3  |-  ( ch  <->  th )
3 syl3anb.3 . . 3  |-  ( ta  <->  et )
41, 2, 33anbi123i 1128 . 2  |-  ( (
ph  /\  ch  /\  ta ) 
<->  ( ps  /\  th  /\  et ) )
5 syl3anb.4 . 2  |-  ( ( ps  /\  th  /\  et )  ->  ze )
64, 5sylbi 119 1  |-  ( (
ph  /\  ch  /\  ta )  ->  ze )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by:  syl3anbr  1214  poxp  5904
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