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	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step22	⊢ ( 𝜑 → 𝜑 )
2		impsingle-step25	⊢ ( ( 𝜑 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )