Theorem adh-minimp-jarr-ax2c-lem3

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

	Ref	Expression
Assertion	adh-minimp-jarr-ax2c-lem3	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → 𝜃 ) ) → ( 𝜑 → 𝜃 ) ) ) → 𝜏 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-jarr-lem2	⊢ ( ( ( 𝜂 → ( ( 𝜁 → 𝜎 ) → ( ( ( 𝜌 → 𝜁 ) → ( 𝜎 → 𝜇 ) ) → ( 𝜁 → 𝜇 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → 𝜃 ) ) → ( 𝜑 → 𝜃 ) ) ) → 𝜏 ) ) → ( ( ( 𝜁 → 𝜎 ) → ( ( ( 𝜌 → 𝜁 ) → ( 𝜎 → 𝜇 ) ) → ( 𝜁 → 𝜇 ) ) ) → 𝜏 ) )
2		adh-minimp-jarr-lem2	⊢ ( ( ( ( 𝜂 → ( ( 𝜁 → 𝜎 ) → ( ( ( 𝜌 → 𝜁 ) → ( 𝜎 → 𝜇 ) ) → ( 𝜁 → 𝜇 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → 𝜃 ) ) → ( 𝜑 → 𝜃 ) ) ) → 𝜏 ) ) → ( ( ( 𝜁 → 𝜎 ) → ( ( ( 𝜌 → 𝜁 ) → ( 𝜎 → 𝜇 ) ) → ( 𝜁 → 𝜇 ) ) ) → 𝜏 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → 𝜃 ) ) → ( 𝜑 → 𝜃 ) ) ) → 𝜏 ) → 𝜏 ) )
3	1 2	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → 𝜑 ) → ( 𝜓 → 𝜃 ) ) → ( 𝜑 → 𝜃 ) ) ) → 𝜏 ) → 𝜏 )