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Theorem syl3anb 1141
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 1135 . 2 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
5 syl3anb.4 . 2 ((𝜓𝜃𝜂) → 𝜁)
64, 5sylbi 209 1 ((𝜑𝜒𝜏) → 𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1068
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 199  df-an 388  df-3an 1070
This theorem is referenced by:  syl3anbr  1142  poxp  7620  infempty  8758  symgsssg  18346  symgfisg  18347  lmodvscl  19363  xrs1mnd  20275  iscnp2  21541  clwwlknccat  27577  slmdvscl  30464  cgr3permute3  32969  cgr3permute1  32970  cgr3permute2  32971  cgr3permute4  32972  cgr3permute5  32973  colinearxfr  32997  grposnOLD  34550  rngunsnply  39114
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