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Theorem syl3anb 1162
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 1156 . 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 1088
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 398  df-3an 1090
This theorem is referenced by:  syl3anbr  1163  poxp  8114  infempty  9502  symgsssg  19335  symgfisg  19336  lmodvscl  20489  xrs1mnd  20983  iscnp2  22743  clwwlknccat  29316  slmdvscl  32359  cgr3permute3  35019  cgr3permute1  35020  cgr3permute2  35021  cgr3permute4  35022  cgr3permute5  35023  colinearxfr  35047  grposnOLD  36750  rngunsnply  41915
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