Theorem adh-minimp-jarr-lem2 48007

Description: Second lemma for the derivation of jarr 107, and indirectly ax-1 6, a commuted form of ax-2 7, and ax-2 7 proper, from adh-minimp 48005 and ax-mp 5. Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minimp-jarr-lem2	⊢ (((𝜑 → 𝜓) → (((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂))) → 𝜁)) → (𝜓 → 𝜁))

Proof of Theorem adh-minimp-jarr-lem2

Step	Hyp	Ref	Expression
1		adh-minimp 48005	. 2 ⊢ (𝜓 → ((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂))))
2		adh-minimp-jarr-imim1-ax2c-lem1 48006	. 2 ⊢ ((𝜓 → ((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂)))) → (((𝜑 → 𝜓) → (((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂))) → 𝜁)) → (𝜓 → 𝜁)))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → (((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂))) → 𝜁)) → (𝜓 → 𝜁))

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-minimp-jarr-ax2c-lem3  48008  adh-minimp-sylsimp  48009