Theorem notnotrALTVD 45571

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 131). 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 45186 is notnotrALTVD 45571 without virtual deductions and was automatically derived from notnotrALTVD 45571. 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 45231	. . . . 5 ⊢ (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2		pm2.21 124	. . . . 5 ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
3	1, 2	e1a 45284	. . . 4 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ 𝜑 → ¬ ¬ ¬ 𝜑)   )
4		con4 114	. . . 4 ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 → 𝜑))
5	3, 4	e1a 45284	. . 3 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑 → 𝜑)   )
6		id 23	. . 3 ⊢ ((¬ ¬ 𝜑 → 𝜑) → (¬ ¬ 𝜑 → 𝜑))
7	5, 1, 6	e11 45345	. 2 ⊢ (    ¬ ¬ 𝜑   ▶   𝜑   )
8	7	in1 45228	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 210  df-vd1 45227

This theorem is referenced by: (None)