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Theorem syl3anl 1233
Description: A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
Hypotheses
Ref Expression
syl3anl.1  |-  ( ph  ->  ps )
syl3anl.2  |-  ( ch 
->  th )
syl3anl.3  |-  ( ta 
->  et )
syl3anl.4  |-  ( ( ( ps  /\  th  /\  et )  /\  ze )  ->  si )
Assertion
Ref Expression
syl3anl  |-  ( ( ( ph  /\  ch  /\ 
ta )  /\  ze )  ->  si )

Proof of Theorem syl3anl
StepHypRef Expression
1 syl3anl.1 . . 3  |-  ( ph  ->  ps )
2 syl3anl.2 . . 3  |-  ( ch 
->  th )
3 syl3anl.3 . . 3  |-  ( ta 
->  et )
41, 2, 33anim123i 1137 . 2  |-  ( (
ph  /\  ch  /\  ta )  ->  ( ps  /\  th 
/\  et ) )
5 syl3anl.4 . 2  |-  ( ( ( ps  /\  th  /\  et )  /\  ze )  ->  si )
64, 5sylan 457 1  |-  ( ( ( ph  /\  ch  /\ 
ta )  /\  ze )  ->  si )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934
This theorem is referenced by:  chlej1  22105  chlej2  22106  atcvatlem  22981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
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