Theorem adh-minim-ax1-ax2-lem3

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

	Ref	Expression
Assertion	adh-minim-ax1-ax2-lem3	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜃 → ( 𝜑 → 𝜒 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem1	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → ( ( 𝜃 → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( 𝜃 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜓 → ( 𝜃 → ( 𝜑 → 𝜒 ) ) ) ) )
2		adh-minim-ax1-ax2-lem2	⊢ ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → ( ( 𝜃 → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( 𝜃 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜓 → ( 𝜃 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜃 → ( 𝜑 → 𝜒 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜃 → ( 𝜑 → 𝜒 ) ) ) )