Theorem adh-minimp-ax2-lem4

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

	Ref	Expression
Assertion	adh-minimp-ax2-lem4	⊢ ( 𝜑 → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-ax2c	⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) )
2		adh-minimp-sylsimp	⊢ ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) ) → ( 𝜑 → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) ) )
3	1 2	ax-mp	⊢ ( 𝜑 → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) )