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Theorem 3imp3i2an 1299
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 3simpb 1079 . . 3 ((𝜑𝜓𝜒) → (𝜑𝜒))
3 3imp3i2an.2 . . 3 ((𝜑𝜒) → 𝜏)
42, 3syl 17 . 2 ((𝜑𝜓𝜒) → 𝜏)
5 3imp3i2an.3 . 2 ((𝜃𝜏) → 𝜂)
61, 4, 5syl2anc 694 1 ((𝜑𝜓𝜒) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054
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 197  df-an 385  df-3an 1056
This theorem is referenced by:  cplgr3v  26387  upgr2pthnlp  26684  frgrreg  27381  eliuniin  39593  eliuniin2  39617
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