# Metamath Proof Explorer

## Theorem con3ALTVD

Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 7 of Section 14 of Margaris p. 60 (which is con3 ). The same proof may also be interpreted to be 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. con3ALT2 is con3ALTVD without virtual deductions and was automatically derived from con3ALTVD . Step i of the User's Proof corresponds to step i of the Fitch-style proof.

 1:: |- (. ( ph -> ps ) ->. ( ph -> ps ) ). 2:: |- (. ( ph -> ps ) ,. -. -. ph ->. -. -. ph ). 3:: |- ( -. -. ph -> ph ) 4:2: |- (. ( ph -> ps ) ,. -. -. ph ->. ph ). 5:1,4: |- (. ( ph -> ps ) ,. -. -. ph ->. ps ). 6:: |- ( ps -> -. -. ps ) 7:6,5: |- (. ( ph -> ps ) ,. -. -. ph ->. -. -. ps ). 8:7: |- (. ( ph -> ps ) ->. ( -. -. ph -> -. -. ps ) ). 9:: |- ( ( -. -. ph -> -. -. ps ) -> ( -. ps -> -. ph ) ) 10:8: |- (. ( ph -> ps ) ->. ( -. ps -> -. ph ) ). qed:10: |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
(Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion con3ALTVD ${⊢}\left({\phi }\to {\psi }\right)\to \left(¬{\psi }\to ¬{\phi }\right)$

### Proof

Step Hyp Ref Expression
1 idn1 ${⊢}\left(\left({\phi }\to {\psi }\right){\to }\left({\phi }\to {\psi }\right)\right)$
2 idn2 ${⊢}\left(\left({\phi }\to {\psi }\right){,}¬¬{\phi }{\to }¬¬{\phi }\right)$
3 notnotr ${⊢}¬¬{\phi }\to {\phi }$
4 2 3 e2 ${⊢}\left(\left({\phi }\to {\psi }\right){,}¬¬{\phi }{\to }{\phi }\right)$
5 id ${⊢}\left({\phi }\to {\psi }\right)\to \left({\phi }\to {\psi }\right)$
6 1 4 5 e12 ${⊢}\left(\left({\phi }\to {\psi }\right){,}¬¬{\phi }{\to }{\psi }\right)$
7 notnot ${⊢}{\psi }\to ¬¬{\psi }$
8 6 7 e2 ${⊢}\left(\left({\phi }\to {\psi }\right){,}¬¬{\phi }{\to }¬¬{\psi }\right)$
9 8 in2 ${⊢}\left(\left({\phi }\to {\psi }\right){\to }\left(¬¬{\phi }\to ¬¬{\psi }\right)\right)$
10 con4 ${⊢}\left(¬¬{\phi }\to ¬¬{\psi }\right)\to \left(¬{\psi }\to ¬{\phi }\right)$
11 9 10 e1a ${⊢}\left(\left({\phi }\to {\psi }\right){\to }\left(¬{\psi }\to ¬{\phi }\right)\right)$
12 11 in1 ${⊢}\left({\phi }\to {\psi }\right)\to \left(¬{\psi }\to ¬{\phi }\right)$