Theorem adh-minim-ax1-ax2-lem1

Description: First lemma for the derivation of ax-1 and ax-2 from adh-minim and ax-mp . Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minim-ax1-ax2-lem1	⊢ ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜂 ) ) → ( 𝜓 → 𝜂 ) ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minim	⊢ ( ( ( 𝜁 → 𝜃 ) → 𝜓 ) → ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) )
2		adh-minim	⊢ ( ( ( ( 𝜁 → 𝜃 ) → 𝜓 ) → ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) ) → ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜂 ) ) → ( 𝜓 → 𝜂 ) ) ) )
3	1 2	ax-mp	⊢ ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜂 ) ) → ( 𝜓 → 𝜂 ) ) )