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Theorem notnotrALTVD 39650
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 125). 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 39237 is notnotrALTVD 39650 without virtual deductions and was automatically derived from notnotrALTVD 39650. 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   ▶   (¬ ¬ 𝜑 → 𝜑)   ) 6:5,1: ⊢ (   ¬ ¬ 𝜑   ▶   𝜑   ) qed:6: ⊢ (¬ ¬ 𝜑 → 𝜑)
(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 39292 . . . . 5 (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2 pm2.21 120 . . . . 5 (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
31, 2e1a 39354 . . . 4 (    ¬ ¬ 𝜑   ▶   𝜑 → ¬ ¬ ¬ 𝜑)   )
4 con4 112 . . . 4 ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑𝜑))
53, 4e1a 39354 . . 3 (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑𝜑)   )
6 id 22 . . 3 ((¬ ¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
75, 1, 6e11 39415 . 2 (    ¬ ¬ 𝜑   ▶   𝜑   )
87in1 39289 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 197  df-vd1 39288 This theorem is referenced by: (None)
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