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Theorem syl3anbr 1267
Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
syl3anbr.1  |-  ( ps  <->  ph )
syl3anbr.2  |-  ( th  <->  ch )
syl3anbr.3  |-  ( et  <->  ta )
syl3anbr.4  |-  ( ( ps  /\  th  /\  et )  ->  ze )
Assertion
Ref Expression
syl3anbr  |-  ( (
ph  /\  ch  /\  ta )  ->  ze )

Proof of Theorem syl3anbr
StepHypRef Expression
1 syl3anbr.1 . . 3  |-  ( ps  <->  ph )
21bicomi 202 . 2  |-  ( ph  <->  ps )
3 syl3anbr.2 . . 3  |-  ( th  <->  ch )
43bicomi 202 . 2  |-  ( ch  <->  th )
5 syl3anbr.3 . . 3  |-  ( et  <->  ta )
65bicomi 202 . 2  |-  ( ta  <->  et )
7 syl3anbr.4 . 2  |-  ( ( ps  /\  th  /\  et )  ->  ze )
82, 4, 6, 7syl3anb 1266 1  |-  ( (
ph  /\  ch  /\  ta )  ->  ze )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968
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 970
This theorem is referenced by:  abvtriv  17266  colinearxfr  29288  paddval  34469
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