Theorem adh-minimp-ax1 47783

Description: Derivation of ax-1 6 from adh-minimp 47778 and ax-mp 5. Polish prefix notation: CpCqp . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem adh-minimp-ax1

Step	Hyp	Ref	Expression
1		adh-minimp-sylsimp 47782	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜓 → 𝜑))
2		adh-minimp-sylsimp 47782	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜑) → (𝜓 → 𝜑)) → (𝜑 → (𝜓 → 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → (𝜓 → 𝜑))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  adh-minimp-idALT  47788  adh-minimp-pm2.43  47789