Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  contrd Structured version   Visualization version   GIF version

Theorem contrd 34229
Description: A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
Hypotheses
Ref Expression
contrd.1 (𝜑 → (¬ 𝜓𝜒))
contrd.2 (𝜑 → (¬ 𝜓 → ¬ 𝜒))
Assertion
Ref Expression
contrd (𝜑𝜓)

Proof of Theorem contrd
StepHypRef Expression
1 contrd.1 . . 3 (𝜑 → (¬ 𝜓𝜒))
2 contrd.2 . . 3 (𝜑 → (¬ 𝜓 → ¬ 𝜒))
31, 2jcad 504 . 2 (𝜑 → (¬ 𝜓 → (𝜒 ∧ ¬ 𝜒)))
4 pm2.24 122 . . . . 5 (𝜒 → (¬ 𝜒𝜓))
54imp 395 . . . 4 ((𝜒 ∧ ¬ 𝜒) → 𝜓)
65imim2i 16 . . 3 ((¬ 𝜓 → (𝜒 ∧ ¬ 𝜒)) → (¬ 𝜓𝜓))
76pm2.18d 127 . 2 ((¬ 𝜓 → (𝜒 ∧ ¬ 𝜒)) → 𝜓)
83, 7syl 17 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384
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 198  df-an 385
This theorem is referenced by:  mpt2bi123f  34300  mptbi12f  34304  ac6s6  34309
  Copyright terms: Public domain W3C validator