Theorem hirstL-ax3

Description: The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of Mendelson p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015) (Proof modification is discouraged.)

	Ref	Expression
Assertion	hirstL-ax3	$$⊢\left(\neg \phi \to \neg \psi \right)\to \left(\left(\neg \phi \to \psi \right)\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm4.64	$$⊢\left(\neg \phi \to \psi \right)\leftrightarrow \left(\phi \vee \psi \right)$$
2		pm4.66	$$⊢\left(\neg \phi \to \neg \psi \right)\leftrightarrow \left(\phi \vee \neg \psi \right)$$
3		pm2.64	$$⊢\left(\phi \vee \psi \right)\to \left(\left(\phi \vee \neg \psi \right)\to \phi \right)$$
4	3	com12	$$⊢\left(\phi \vee \neg \psi \right)\to \left(\left(\phi \vee \psi \right)\to \phi \right)$$
5	2 4	sylbi	$$⊢\left(\neg \phi \to \neg \psi \right)\to \left(\left(\phi \vee \psi \right)\to \phi \right)$$
6	1 5	biimtrid	$$⊢\left(\neg \phi \to \neg \psi \right)\to \left(\left(\neg \phi \to \psi \right)\to \phi \right)$$