Theorem minimp-ax1 1631

Description: Derivation of ax-1 6 from ax-mp 5 and minimp 1629. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem minimp-ax1

Step	Hyp	Ref	Expression
1		minimp-syllsimp 1630	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜓 → 𝜑))
2		minimp-syllsimp 1630	. 2 ⊢ ((((𝜑 → 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

This theorem is referenced by:  minimp-pm2.43  1634