Theorem peirce 76

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 3 through ax-3 5. A curious fact about this theorem is that it requires ax-3 5 for its proof even though the result has no negations in it.

Assertion

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

Proof of Theorem peirce

Step	Hyp	Ref	Expression
1		pm2.21 71	. . 3 |- (-. ph -> (ph -> ps))
2	1	syl4 19	. 2 |- (((ph -> ps) -> ph) -> (-. ph -> ph))
3		pm2.18 75	. 2 |- ((-. ph -> ph) -> ph)
4	2, 3	syl 12	1 |- (((ph -> ps) -> ph) -> ph)

Syntax hints:  -. wn 1   -> wi 2

This theorem is referenced by:  looinv 77  exmoeu 1039

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

metamath.org