Theorem peirce

Description: Peirce's axiom. A non-intuitionistic implication-only statement. Added to intuitionistic (implicational) propositional calculus, it gives classical (implicational) propositional calculus. For another non-intuitionistic positive statement, see curryax . When F. is substituted for ps , then this becomes the Clavius law pm2.18 . (Contributed by NM, 29-Dec-1992) (Proof shortened by Wolf Lammen, 9-Oct-2012)

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

Proof

Step	Hyp	Ref	Expression
1		simplim	$$⊢\neg \left(\phi \to \psi \right)\to \phi$$
2		id	$$⊢\phi \to \phi$$
3	1 2	ja	$$⊢\left(\left(\phi \to \psi \right)\to \phi \right)\to \phi$$