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Theorem peirce 188
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 notable 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, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
Ref Expression
peirce (((𝜑𝜓) → 𝜑) → 𝜑)

Proof of Theorem peirce
StepHypRef Expression
1 simplim 158 . 2 (¬ (𝜑𝜓) → 𝜑)
2 id 22 . 2 (𝜑𝜑)
31, 2ja 168 1 (((𝜑𝜓) → 𝜑) → 𝜑)
Colors of variables: wff setvar class
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  189  tbw-ax3  1615  tb-ax3  31190  bj-peircecurry  31325  bj-peircei  31333
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