MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl3anl1 Structured version   Unicode version

Theorem syl3anl1 1266
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
syl3anl1.1  |-  ( ph  ->  ps )
syl3anl1.2  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta )  ->  et )
Assertion
Ref Expression
syl3anl1  |-  ( ( ( ph  /\  ch  /\ 
th )  /\  ta )  ->  et )

Proof of Theorem syl3anl1
StepHypRef Expression
1 syl3anl1.1 . . 3  |-  ( ph  ->  ps )
213anim1i 1174 . 2  |-  ( (
ph  /\  ch  /\  th )  ->  ( ps  /\  ch  /\  th ) )
3 syl3anl1.2 . 2  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ta )  ->  et )
42, 3sylan 471 1  |-  ( ( ( ph  /\  ch  /\ 
th )  /\  ta )  ->  et )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967
This theorem is referenced by:  suprzcl  10720  latjcom  15228  latmcom  15244  lgsdinn0  22678  crngohomfo  28804  dalem53  33367
  Copyright terms: Public domain W3C validator