Theorem impsingle-ax1

Description: Derivation of impsingle-ax1 ( ax-1 ) from ax-mp and impsingle . Lukasiewicz was used to construct this proof. Every formula corresponding to a detachment step was converted to its corresponding Metamath formula. mmj2 was used to unify each formula using ax-mp , which in turn produced this proof. With extremely high confidence, this result shows that the Lukasiewicz proof of ax-1 (step 27) is correct and that Metamath correctly verifies the proof. The same comments apply to the proofs that follow this one. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	impsingle-ax1	$$⊢\phi \to \left(\psi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		impsingle-step8	$$⊢\left(\left(\chi \to \psi \right)\to \phi \right)\to \left(\psi \to \phi \right)$$
2		impsingle-step8	$$⊢\left(\left(\left(\chi \to \psi \right)\to \phi \right)\to \left(\psi \to \phi \right)\right)\to \left(\phi \to \left(\psi \to \phi \right)\right)$$
3	1 2	ax-mp	$$⊢\phi \to \left(\psi \to \phi \right)$$