Theorem adh-minimp-ax2

Description: Derivation of ax-2 from adh-minimp and ax-mp . Polish prefix notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021) (Revised by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-ax2-lem4	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) )
2		adh-minimp-ax2c	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) )
3		adh-minimp-ax2-lem4	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) )
5	1 4	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )