Theorem peirce 172

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 6 through ax-3 8. A curious fact about this theorem is that it requires ax-3 8 for its proof even though the result has no negation connectives in it. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)

Assertion

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

Proof of Theorem peirce

Step	Hyp	Ref	Expression
1		simplim 143	. 2 ⊢ (¬ (φ → ψ) → φ)
2		id 19	. 2 ⊢ (φ → φ)
3	1, 2	ja 153	1 ⊢ (((φ → ψ) → φ) → φ)

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  looinv  174  tbw-ax3  1467  exmoeu  2246