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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 5 through ax-3 7. A curious fact about this theorem is that it requires ax-3 7 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 (((φψ) → φ) → φ)

Proof of Theorem peirce
StepHypRef Expression
1 simplim 143 . 2 (¬ (φψ) → φ)
2 id 19 . 2 (φφ)
31, 2ja 153 1 (((φψ) → φ) → φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem is referenced by:  looinv  174  tbw-ax3  1467  exmoeu  2246
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