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Theorem syl3anbr 1367
 Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
syl3anbr.1 (𝜓𝜑)
syl3anbr.2 (𝜃𝜒)
syl3anbr.3 (𝜂𝜏)
syl3anbr.4 ((𝜓𝜃𝜂) → 𝜁)
Assertion
Ref Expression
syl3anbr ((𝜑𝜒𝜏) → 𝜁)

Proof of Theorem syl3anbr
StepHypRef Expression
1 syl3anbr.1 . . 3 (𝜓𝜑)
21bicomi 214 . 2 (𝜑𝜓)
3 syl3anbr.2 . . 3 (𝜃𝜒)
43bicomi 214 . 2 (𝜒𝜃)
5 syl3anbr.3 . . 3 (𝜂𝜏)
65bicomi 214 . 2 (𝜏𝜂)
7 syl3anbr.4 . 2 ((𝜓𝜃𝜂) → 𝜁)
82, 4, 6, 7syl3anb 1366 1 ((𝜑𝜒𝜏) → 𝜁)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1036 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 197  df-an 386  df-3an 1038 This theorem is referenced by:  abvtriv  18781  colinearxfr  31877  paddval  34603
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