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Theorem syl3anbr 1228
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 194 . 2  |-  ( ph  <->  ps )
3 syl3anbr.2 . . 3  |-  ( th  <->  ch )
43bicomi 194 . 2  |-  ( ch  <->  th )
5 syl3anbr.3 . . 3  |-  ( et  <->  ta )
65bicomi 194 . 2  |-  ( ta  <->  et )
7 syl3anbr.4 . 2  |-  ( ( ps  /\  th  /\  et )  ->  ze )
82, 4, 6, 7syl3anb 1227 1  |-  ( (
ph  /\  ch  /\  ta )  ->  ze )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936
This theorem is referenced by:  abvtriv  15929  colinearxfr  26009  paddval  30595
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 178  df-an 361  df-3an 938
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