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.

	Ref	Expression
Assertion	con3ALTVD	$$⊢\left(\phi \to \psi \right)\to \left(\neg \psi \to \neg \phi \right)$$

Proof

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