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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	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		simplim	⊢ ( ¬ ( 𝜑 → 𝜓 ) → 𝜑 )
2		id	⊢ ( 𝜑 → 𝜑 )
3	1 2	ja	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )