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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 
|- ( ( ( ph -> ps ) -> ph ) -> ph )

Proof

Step Hyp Ref Expression
1 simplim 
 |-  ( -. ( ph -> ps ) -> ph )
2 id 
 |-  ( ph -> ph )
3 1 2 ja 
 |-  ( ( ( ph -> ps ) -> ph ) -> ph )