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Theorem syl3anb 1153
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 1147 . 2 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
5 syl3anb.4 . 2 ((𝜓𝜃𝜂) → 𝜁)
64, 5sylbi 218 1 ((𝜑𝜒𝜏) → 𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079
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 208  df-an 397  df-3an 1081
This theorem is referenced by:  syl3anbr  1154  poxp  7811  infempty  8959  symgsssg  18524  symgfisg  18525  lmodvscl  19580  xrs1mnd  20511  iscnp2  21775  clwwlknccat  27769  slmdvscl  30769  cgr3permute3  33405  cgr3permute1  33406  cgr3permute2  33407  cgr3permute4  33408  cgr3permute5  33409  colinearxfr  33433  grposnOLD  35041  rngunsnply  39651
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