Theorem adh-minim-ax1-ax2-lem2 48017

Description: Second lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 48015 and ax-mp 5. Polish prefix notation: CCpCCqCCrCpsCrstCpt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minim-ax1-ax2-lem2	⊢ ((𝜑 → ((𝜓 → ((𝜒 → (𝜑 → 𝜃)) → (𝜒 → 𝜃))) → 𝜏)) → (𝜑 → 𝜏))

Proof of Theorem adh-minim-ax1-ax2-lem2

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem1 48016	. 2 ⊢ (𝜂 → ((𝜁 → ((𝜎 → ((𝜌 → (𝜁 → 𝜇)) → (𝜌 → 𝜇))) → 𝜆)) → (𝜁 → 𝜆)))
2		adh-minim-ax1-ax2-lem1 48016	. 2 ⊢ ((𝜂 → ((𝜁 → ((𝜎 → ((𝜌 → (𝜁 → 𝜇)) → (𝜌 → 𝜇))) → 𝜆)) → (𝜁 → 𝜆))) → ((𝜑 → ((𝜓 → ((𝜒 → (𝜑 → 𝜃)) → (𝜒 → 𝜃))) → 𝜏)) → (𝜑 → 𝜏)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → ((𝜓 → ((𝜒 → (𝜑 → 𝜃)) → (𝜒 → 𝜃))) → 𝜏)) → (𝜑 → 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-ax1-ax2-lem3  48018  adh-minim-ax1-ax2-lem4  48019