Theorem adh-minim-ax1-ax2-lem1 43313

Description: First lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43312 and ax-mp 5. Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minim-ax1-ax2-lem1	⊢ (𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜂)) → (𝜓 → 𝜂)))

Proof of Theorem adh-minim-ax1-ax2-lem1

Step	Hyp	Ref	Expression
1		adh-minim 43312	. 2 ⊢ (((𝜁 → 𝜃) → 𝜓) → (𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))))
2		adh-minim 43312	. 2 ⊢ ((((𝜁 → 𝜃) → 𝜓) → (𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏)))) → (𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜂)) → (𝜓 → 𝜂))))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜂)) → (𝜓 → 𝜂)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  adh-minim-ax1-ax2-lem2  43314  adh-minim-ax1-ax2-lem3  43315  adh-minim-ax1  43317