Theorem impsingle-ax1 1630

Description: Derivation of impsingle-ax1 (ax-1 6) from ax-mp 5 and impsingle 1627. 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 5, which in turn produced this proof. With extremely high confidence, this result shows that the Lukasiewicz proof of ax-1 6 (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.)

Assertion

Ref	Expression
impsingle-ax1	⊢ (𝜑 → (𝜓 → 𝜑))

Proof of Theorem impsingle-ax1

Step	Hyp	Ref	Expression
1		impsingle-step8 1629	. 2 ⊢ (((𝜒 → 𝜓) → 𝜑) → (𝜓 → 𝜑))
2		impsingle-step8 1629	. 2 ⊢ ((((𝜒 → 𝜓) → 𝜑) → (𝜓 → 𝜑)) → (𝜑 → (𝜓 → 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → (𝜓 → 𝜑))

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: (None)