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

Theorem nic-mpALT 1674
 Description: A direct proof of nic-mp 1673. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nic-jmin 𝜑
nic-jmaj (𝜑 ⊼ (𝜒𝜓))
Assertion
Ref Expression
nic-mpALT 𝜓

Proof of Theorem nic-mpALT
StepHypRef Expression
1 nic-jmin . 2 𝜑
2 nic-jmaj . . . . 5 (𝜑 ⊼ (𝜒𝜓))
3 df-nan 1483 . . . . . 6 ((𝜑 ⊼ (𝜒𝜓)) ↔ ¬ (𝜑 ∧ (𝜒𝜓)))
4 df-nan 1483 . . . . . . 7 ((𝜒𝜓) ↔ ¬ (𝜒𝜓))
54anbi2i 625 . . . . . 6 ((𝜑 ∧ (𝜒𝜓)) ↔ (𝜑 ∧ ¬ (𝜒𝜓)))
63, 5xchbinx 337 . . . . 5 ((𝜑 ⊼ (𝜒𝜓)) ↔ ¬ (𝜑 ∧ ¬ (𝜒𝜓)))
72, 6mpbi 233 . . . 4 ¬ (𝜑 ∧ ¬ (𝜒𝜓))
8 iman 405 . . . 4 ((𝜑 → (𝜒𝜓)) ↔ ¬ (𝜑 ∧ ¬ (𝜒𝜓)))
97, 8mpbir 234 . . 3 (𝜑 → (𝜒𝜓))
109simprd 499 . 2 (𝜑𝜓)
111, 10ax-mp 5 1 𝜓
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ⊼ wnan 1482 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 210  df-an 400  df-nan 1483 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator