# Metamath Proof Explorer

## Theorem notnotrALTVD

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 ). 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 is notnotrALTVD without virtual deductions and was automatically derived from notnotrALTVD . Step i of the User's Proof corresponds to step i of the Fitch-style proof.

 1:: |- (. -. -. ph ->. -. -. ph ). 2:: |- ( -. -. ph -> ( -. ph -> -. -. -. ph ) ) 3:1: |- (. -. -. ph ->. ( -. ph -> -. -. -. ph ) ). 4:: |- ( ( -. ph -> -. -. -. ph ) -> ( -. -. ph -> ph ) ) 5:3: |- (. -. -. ph ->. ( -. -. ph -> ph ) ). 6:5,1: |- (. -. -. ph ->. ph ). qed:6: |- ( -. -. ph -> ph )
(Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion notnotrALTVD ${⊢}¬¬{\phi }\to {\phi }$

### Proof

Step Hyp Ref Expression
1 idn1 ${⊢}\left(¬¬{\phi }{\to }¬¬{\phi }\right)$
2 pm2.21 ${⊢}¬¬{\phi }\to \left(¬{\phi }\to ¬¬¬{\phi }\right)$
3 1 2 e1a ${⊢}\left(¬¬{\phi }{\to }\left(¬{\phi }\to ¬¬¬{\phi }\right)\right)$
4 con4 ${⊢}\left(¬{\phi }\to ¬¬¬{\phi }\right)\to \left(¬¬{\phi }\to {\phi }\right)$
5 3 4 e1a ${⊢}\left(¬¬{\phi }{\to }\left(¬¬{\phi }\to {\phi }\right)\right)$
6 id ${⊢}\left(¬¬{\phi }\to {\phi }\right)\to \left(¬¬{\phi }\to {\phi }\right)$
7 5 1 6 e11 ${⊢}\left(¬¬{\phi }{\to }{\phi }\right)$
8 7 in1 ${⊢}¬¬{\phi }\to {\phi }$