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Theorem peirce 204
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 889. When is substituted for 𝜓, then this becomes the Clavius law pm2.18 128. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
Assertion
Ref Expression
peirce (((𝜑𝜓) → 𝜑) → 𝜑)

Proof of Theorem peirce
StepHypRef Expression
1 simplim 168 . 2 (¬ (𝜑𝜓) → 𝜑)
2 id 22 . 2 (𝜑𝜑)
31, 2ja 187 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  205  impsingle  1621  tarski-bernays-ax2  1634  tbw-ax3  1696  tb-ax3  33721  bj-peircestab  33876  bj-peircecurry  33881  bj-peircei  33889
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