Theorem peirce 76

Description: Peirce's axiom. This odd-looking theorem is the "difference" between an intuitionistic system of propositional calculus and a classical system and is not accepted by intuitionists. When Peirce's axiom is added to an intuitionistic system, the system becomes equivalent to our classical system ax-1 3 through ax-3 5. A curious fact about this theorem is that it requires ax-3 5 for its proof even though the result has no negations in it.

Assertion

Ref	Expression
peirce	⊢ (((φ → ψ) → φ) → φ)

Proof of Theorem peirce

Step	Hyp	Ref	Expression
1		pm2.21 71	. . 3 ⊢ (¬ φ → (φ → ψ))
2	1	syl4 19	. 2 ⊢ (((φ → ψ) → φ) → (¬ φ → φ))
3		pm2.18 75	. 2 ⊢ ((¬ φ → φ) → φ)
4	2, 3	syl 12	1 ⊢ (((φ → ψ) → φ) → φ)

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  looinv 77  exmoeu 1039

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

metamath.org