MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mp3anl1 Structured version   Visualization version   GIF version

Theorem mp3anl1 1448
Description: An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
mp3anl1.1 𝜑
mp3anl1.2 (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
mp3anl1 (((𝜓𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem mp3anl1
StepHypRef Expression
1 mp3anl1.1 . . 3 𝜑
2 mp3anl1.2 . . . 4 (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
32ex 413 . . 3 ((𝜑𝜓𝜒) → (𝜃𝜏))
41, 3mp3an1 1441 . 2 ((𝜓𝜒) → (𝜃𝜏))
54imp 407 1 (((𝜓𝜒) ∧ 𝜃) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081
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 208  df-an 397  df-3an 1083
This theorem is referenced by:  mp3anr1  1451  facavg  13651  iddvds  15613  isprm7  16042  blometi  28494  mdslmd3i  30023  atcvat2i  30078  chirredlem3  30083  mdsymlem1  30094
  Copyright terms: Public domain W3C validator