Theorem peirce 193

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.)

Assertion

Ref	Expression
peirce	|- ( ( ( ph -> ps ) -> ph ) -> ph )

Proof of Theorem peirce

Step	Hyp	Ref	Expression
1		simplim 163	. 2 |- ( -. ( ph -> ps ) -> ph )
2		id 22	. 2 |- ( ph -> ph )
3	1, 2	ja 173	1 |- ( ( ( ph -> ps ) -> ph ) -> ph )

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  194  tbw-ax3  1627  tb-ax3  32380  bj-peircecurry  32545  bj-peircei  32553