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Theorem notnotrALTVD 43289
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 130). 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 42903 is notnotrALTVD 43289 without virtual deductions and was automatically derived from notnotrALTVD 43289. 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 42948 . . . . 5 (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2 pm2.21 123 . . . . 5 (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
31, 2e1a 43001 . . . 4 (    ¬ ¬ 𝜑   ▶   𝜑 → ¬ ¬ ¬ 𝜑)   )
4 con4 113 . . . 4 ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑𝜑))
53, 4e1a 43001 . . 3 (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑𝜑)   )
6 id 22 . . 3 ((¬ ¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
75, 1, 6e11 43062 . 2 (    ¬ ¬ 𝜑   ▶   𝜑   )
87in1 42945 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 206  df-vd1 42944
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator