Theorem adh-minim-ax1-ax2-lem2

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

	Ref	Expression
Assertion	adh-minim-ax1-ax2-lem2	⊢ ( ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( 𝜑 → 𝜃 ) ) → ( 𝜒 → 𝜃 ) ) ) → 𝜏 ) ) → ( 𝜑 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem1	⊢ ( 𝜂 → ( ( 𝜁 → ( ( 𝜎 → ( ( 𝜌 → ( 𝜁 → 𝜇 ) ) → ( 𝜌 → 𝜇 ) ) ) → 𝜆 ) ) → ( 𝜁 → 𝜆 ) ) )
2		adh-minim-ax1-ax2-lem1	⊢ ( ( 𝜂 → ( ( 𝜁 → ( ( 𝜎 → ( ( 𝜌 → ( 𝜁 → 𝜇 ) ) → ( 𝜌 → 𝜇 ) ) ) → 𝜆 ) ) → ( 𝜁 → 𝜆 ) ) ) → ( ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( 𝜑 → 𝜃 ) ) → ( 𝜒 → 𝜃 ) ) ) → 𝜏 ) ) → ( 𝜑 → 𝜏 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( 𝜑 → 𝜃 ) ) → ( 𝜒 → 𝜃 ) ) ) → 𝜏 ) ) → ( 𝜑 → 𝜏 ) )