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Theorem 3imp3i2an 1346
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 1132 . 2 ((𝜑𝜓𝜒) → 𝜏)
4 3imp3i2an.3 . 2 ((𝜃𝜏) → 𝜂)
51, 3, 4syl2anc 585 1 ((𝜑𝜓𝜒) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  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:  focofo  6819  ordunel  7815  naddel1  8686  distrlem5pr  11022  divmul  11875  modmulnn  13854  moddi  13904  repswpfx  14735  shftval2  15022  pcgcd  16811  gsumccat  18722  qussub  19070  gsumdixp  20131  lspun  20598  evlslem4  21637  ordtcld3  22703  sleadd1im  27470  fusgrfisstep  28586  cplgr3v  28692  upgr2pthnlp  28989  frgrreg  29647  eliuniin  43788  eliuniin2  43809  disjinfi  43891
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