Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mp3an2ani Structured version   Visualization version   GIF version

Theorem mp3an2ani 1423
 Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
Hypotheses
Ref Expression
mp3an2ani.1 𝜑
mp3an2ani.2 (𝜓𝜒)
mp3an2ani.3 ((𝜓𝜃) → 𝜏)
mp3an2ani.4 ((𝜑𝜒𝜏) → 𝜂)
Assertion
Ref Expression
mp3an2ani ((𝜓𝜃) → 𝜂)

Proof of Theorem mp3an2ani
StepHypRef Expression
1 mp3an2ani.1 . . 3 𝜑
2 mp3an2ani.2 . . 3 (𝜓𝜒)
3 mp3an2ani.3 . . 3 ((𝜓𝜃) → 𝜏)
4 mp3an2ani.4 . . 3 ((𝜑𝜒𝜏) → 𝜂)
51, 2, 3, 4mp3an3an 1422 . 2 ((𝜓 ∧ (𝜓𝜃)) → 𝜂)
65anabss5 853 1 ((𝜓𝜃) → 𝜂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031 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 196  df-an 385  df-3an 1033 This theorem is referenced by:  2lgsoddprmlem2  24934  isosctrlem1ALT  38192  odz2prm2pw  40013  lighneallem4  40065
 Copyright terms: Public domain W3C validator