Theorem notnotrALTVD 39151

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 38735 is notnotrALTVD 39151 without virtual deductions and was automatically derived from notnotrALTVD 39151. 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	⊢ (¬ ¬ 𝜑 → 𝜑)

Proof of Theorem notnotrALTVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . 5 ⊢ (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2		pm2.21 120	. . . . 5 ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
3	1, 2	e1a 38852	. . . 4 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ 𝜑 → ¬ ¬ ¬ 𝜑)   )
4		con4 112	. . . 4 ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 → 𝜑))
5	3, 4	e1a 38852	. . 3 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑 → 𝜑)   )
6		id 22	. . 3 ⊢ ((¬ ¬ 𝜑 → 𝜑) → (¬ ¬ 𝜑 → 𝜑))
7	5, 1, 6	e11 38913	. 2 ⊢ (    ¬ ¬ 𝜑   ▶   𝜑   )
8	7	in1 38787	1 ⊢ (¬ ¬ 𝜑 → 𝜑)

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 38786

This theorem is referenced by: (None)