Theorem adh-minim-ax1-ax2-lem4 43316

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

Assertion

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

Proof of Theorem adh-minim-ax1-ax2-lem4

Step	Hyp	Ref	Expression
1		adh-minim 43312	. 2 ⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜂 → ((𝜁 → (((𝜑 → 𝜓) → 𝜒) → 𝜎)) → (𝜁 → 𝜎))) → ((𝜓 → (𝜒 → 𝜃)) → (𝜓 → 𝜃))))
2		adh-minim-ax1-ax2-lem2 43314	. 2 ⊢ ((((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 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜓 → (𝜒 → 𝜃)) → (𝜓 → 𝜃)))

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  43317  adh-minim-ax2-lem5  43318  adh-minim-ax2-lem6  43319  adh-minim-ax2c  43320