Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  notnotrALTVD Structured version   Visualization version   GIF version

Theorem notnotrALTVD 37976
Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 5 of Section 14 of [Margaris] p. 59 (which is notnotr 123). The same proof may also be interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. notnotrALT 37559 is notnotrALTVD 37976 without virtual deductions and was automatically derived from notnotrALTVD 37976. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
1:: (   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜑   )
2:: (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
3:1: (   ¬ ¬ 𝜑   ▶   𝜑 → ¬ ¬ ¬ 𝜑)   )
4:: ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 𝜑))
5:3: (   ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑𝜑)   )
6:5,1: (   ¬ ¬ 𝜑   ▶   𝜑   )
qed:6: (¬ ¬ 𝜑𝜑)
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
notnotrALTVD (¬ ¬ 𝜑𝜑)

Proof of Theorem notnotrALTVD
StepHypRef Expression
1 idn1 37614 . . . . 5 (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2 pm2.21 118 . . . . 5 (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
31, 2e1a 37676 . . . 4 (    ¬ ¬ 𝜑   ▶   𝜑 → ¬ ¬ ¬ 𝜑)   )
4 con4 110 . . . 4 ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑𝜑))
53, 4e1a 37676 . . 3 (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑𝜑)   )
6 id 22 . . 3 ((¬ ¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
75, 1, 6e11 37737 . 2 (    ¬ ¬ 𝜑   ▶   𝜑   )
87in1 37611 1 (¬ ¬ 𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4
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-vd1 37610
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator