Theorem notnotrALTVD 38173

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 124). 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 37756 is notnotrALTVD 38173 without virtual deductions and was automatically derived from notnotrALTVD 38173. 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 37811	. . . . 5 ⊢ (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2		pm2.21 119	. . . . 5 ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
3	1, 2	e1a 37873	. . . 4 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ 𝜑 → ¬ ¬ ¬ 𝜑)   )
4		con4 111	. . . 4 ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 → 𝜑))
5	3, 4	e1a 37873	. . 3 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑 → 𝜑)   )
6		id 22	. . 3 ⊢ ((¬ ¬ 𝜑 → 𝜑) → (¬ ¬ 𝜑 → 𝜑))
7	5, 1, 6	e11 37934	. 2 ⊢ (    ¬ ¬ 𝜑   ▶   𝜑   )
8	7	in1 37808	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 196  df-vd1 37807

This theorem is referenced by: (None)