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Theorem mp3anl3 1411
Description: An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
mp3anl3.1 𝜒
mp3anl3.2 (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
mp3anl3 (((𝜑𝜓) ∧ 𝜃) → 𝜏)

Proof of Theorem mp3anl3
StepHypRef Expression
1 mp3anl3.1 . . 3 𝜒
2 mp3anl3.2 . . . 4 (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
32ex 448 . . 3 ((𝜑𝜓𝜒) → (𝜃𝜏))
41, 3mp3an3 1404 . 2 ((𝜑𝜓) → (𝜃𝜏))
54imp 443 1 (((𝜑𝜓) ∧ 𝜃) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030
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 195  df-an 384  df-3an 1032
This theorem is referenced by:  mp3anr3  1414  ioombl  23057  nmopadjlem  28138  nmopcoadji  28150  atcvat3i  28445
  Copyright terms: Public domain W3C validator