Theorem adh-minim-ax1-ax2-lem4

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

	Ref	Expression
Assertion	adh-minim-ax1-ax2-lem4	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) → ( 𝜓 → 𝜃 ) ) )

Proof

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