Theorem notnotrALTVD 46086

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 45701 is notnotrALTVD 46086 without virtual deductions and was automatically derived from notnotrALTVD 46086. 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 45746	. . . . 5 ⊢ (    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
2		pm2.21 124	. . . . 5 ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
3	1, 2	e1a 45799	. . . 4 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ 𝜑 → ¬ ¬ ¬ 𝜑)   )
4		con4 114	. . . 4 ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 → 𝜑))
5	3, 4	e1a 45799	. . 3 ⊢ (    ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑 → 𝜑)   )
6		id 23	. . 3 ⊢ ((¬ ¬ 𝜑 → 𝜑) → (¬ ¬ 𝜑 → 𝜑))
7	5, 1, 6	e11 45860	. 2 ⊢ (    ¬ ¬ 𝜑   ▶   𝜑   )
8	7	in1 45743	1 ⊢ (¬ ¬ 𝜑 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-vd1 45742

This theorem is used by: (None)