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Theorem syl3anb 1361
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 1244 . 2 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
5 syl3anb.4 . 2 ((𝜓𝜃𝜂) → 𝜁)
64, 5sylbi 206 1 ((𝜑𝜒𝜏) → 𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  w3a 1031
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 196  df-an 385  df-3an 1033
This theorem is referenced by:  syl3anbr  1362  poxp  7176  infempty  8295  symgsssg  17710  symgfisg  17711  lmodvscl  18703  xrs1mnd  19603  iscnp2  20853  nb3grapr  25982  slmdvscl  29098  cgr3permute3  31324  cgr3permute1  31325  cgr3permute2  31326  cgr3permute4  31327  cgr3permute5  31328  colinearxfr  31352  grposnOLD  32851  rngunsnply  36762
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