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Theorem syl3anb 1161
Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
Hypotheses
Ref Expression
syl3anb.1 (𝜑𝜓)
syl3anb.2 (𝜒𝜃)
syl3anb.3 (𝜏𝜂)
syl3anb.4 ((𝜓𝜃𝜂) → 𝜁)
Assertion
Ref Expression
syl3anb ((𝜑𝜒𝜏) → 𝜁)

Proof of Theorem syl3anb
StepHypRef Expression
1 syl3anb.1 . . 3 (𝜑𝜓)
2 syl3anb.2 . . 3 (𝜒𝜃)
3 syl3anb.3 . . 3 (𝜏𝜂)
41, 2, 33anbi123i 1155 . 2 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
5 syl3anb.4 . 2 ((𝜓𝜃𝜂) → 𝜁)
64, 5sylbi 216 1 ((𝜑𝜒𝜏) → 𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087
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 206  df-an 397  df-3an 1089
This theorem is referenced by:  syl3anbr  1162  poxp  8113  infempty  9501  symgsssg  19334  symgfisg  19335  lmodvscl  20488  xrs1mnd  20982  iscnp2  22742  clwwlknccat  29313  slmdvscl  32354  cgr3permute3  35014  cgr3permute1  35015  cgr3permute2  35016  cgr3permute4  35017  cgr3permute5  35018  colinearxfr  35042  grposnOLD  36745  rngunsnply  41905
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