Theorem peirce 201

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 891. 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	⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

Proof of Theorem peirce

Step	Hyp	Ref	Expression
1		simplim 167	. 2 ⊢ (¬ (𝜑 → 𝜓) → 𝜑)
2		id 22	. 2 ⊢ (𝜑 → 𝜑)
3	1, 2	ja 186	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  202  impsingle  1630  tarski-bernays-ax2  1643  tbw-ax3  1705  tb-ax3  34574  bj-peircestab  34733  bj-peircecurry  34738  bj-peircei  34746