Theorem adh-minimp-jarr-lem2

Description: Second lemma for the derivation of jarr , and indirectly ax-1 , a commuted form of ax-2 , and ax-2 proper, from adh-minimp and ax-mp . Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minimp-jarr-lem2	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜃 ) → ( ( ( 𝜏 → 𝜒 ) → ( 𝜃 → 𝜂 ) ) → ( 𝜒 → 𝜂 ) ) ) → 𝜁 ) ) → ( 𝜓 → 𝜁 ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minimp	⊢ ( 𝜓 → ( ( 𝜒 → 𝜃 ) → ( ( ( 𝜏 → 𝜒 ) → ( 𝜃 → 𝜂 ) ) → ( 𝜒 → 𝜂 ) ) ) )
2		adh-minimp-jarr-imim1-ax2c-lem1	⊢ ( ( 𝜓 → ( ( 𝜒 → 𝜃 ) → ( ( ( 𝜏 → 𝜒 ) → ( 𝜃 → 𝜂 ) ) → ( 𝜒 → 𝜂 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜃 ) → ( ( ( 𝜏 → 𝜒 ) → ( 𝜃 → 𝜂 ) ) → ( 𝜒 → 𝜂 ) ) ) → 𝜁 ) ) → ( 𝜓 → 𝜁 ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜃 ) → ( ( ( 𝜏 → 𝜒 ) → ( 𝜃 → 𝜂 ) ) → ( 𝜒 → 𝜂 ) ) ) → 𝜁 ) ) → ( 𝜓 → 𝜁 ) )