Theorem adh-minim-ax2-lem5 48021

Description: Fifth lemma for the derivation of ax-2 7 from adh-minim 48015 and ax-mp 5. Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minim-ax2-lem5	⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))))

Proof of Theorem adh-minim-ax2-lem5

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem4 48019	. 2 ⊢ (((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏)))
2		adh-minim-ax1 48020	. 2 ⊢ ((((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))) → (𝜑 → (((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏)))))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  adh-minim-ax2-lem6  48022  adh-minim-ax2c  48023