Metamath Proof Explorer

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 φ ψ φ φ


Step Hyp Ref Expression
1 simplim ¬ φ ψ φ
2 id φ φ
3 1 2 ja φ ψ φ φ