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Theorem 3imp3i2an 1407
Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2022.)
Hypotheses
Ref Expression
3imp3i2an.1 ((𝜑𝜓𝜒) → 𝜃)
3imp3i2an.2 ((𝜑𝜒) → 𝜏)
3imp3i2an.3 ((𝜃𝜏) → 𝜂)
Assertion
Ref Expression
3imp3i2an ((𝜑𝜓𝜒) → 𝜂)

Proof of Theorem 3imp3i2an
StepHypRef Expression
1 3imp3i2an.1 . 2 ((𝜑𝜓𝜒) → 𝜃)
2 3imp3i2an.2 . . 3 ((𝜑𝜒) → 𝜏)
323adant2 1122 . 2 ((𝜑𝜓𝜒) → 𝜏)
4 3imp3i2an.3 . 2 ((𝜃𝜏) → 𝜂)
51, 3, 4syl2anc 579 1 ((𝜑𝜓𝜒) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071
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 387  df-3an 1073
This theorem is referenced by:  ordunel  7307  distrlem5pr  10186  divmul  11039  modmulnn  13012  moddi  13062  shftval2  14228  pcgcd  15997  gsumccat  17775  qussub  18049  gsumdixp  19007  lspun  19393  evlslem4  19915  scmatrngiso  20758  ordtcld3  21422  cplgr3v  26800  upgr2pthnlp  27101  frgrreg  27843  eliuniin  40224  eliuniin2  40246
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