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Theorem notnotrALTVD 41542
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 132). 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 41156 is notnotrALTVD 41542 without virtual deductions and was automatically derived from notnotrALTVD 41542. 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 41201 . . . . 5 (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2 pm2.21 123 . . . . 5 (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
31, 2e1a 41254 . . . 4 (    ¬ ¬ 𝜑   ▶   𝜑 → ¬ ¬ ¬ 𝜑)   )
4 con4 113 . . . 4 ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑𝜑))
53, 4e1a 41254 . . 3 (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑𝜑)   )
6 id 22 . . 3 ((¬ ¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
75, 1, 6e11 41315 . 2 (    ¬ ¬ 𝜑   ▶   𝜑   )
87in1 41198 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 210  df-vd1 41197
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator