Theorem impsingle-peirce

Description: Derivation of impsingle-peirce ( peirce ) from ax-mp and impsingle . It is step 28 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	impsingle-peirce	$$⊢\left(\left(\phi \to \psi \right)\to \phi \right)\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		impsingle-step22	$$⊢\phi \to \phi$$
2		impsingle-step25	$$⊢\left(\phi \to \phi \right)\to \left(\left(\left(\phi \to \psi \right)\to \phi \right)\to \phi \right)$$
3	1 2	ax-mp	$$⊢\left(\left(\phi \to \psi \right)\to \phi \right)\to \phi$$