Theorem adh-minimp-jarr-imim1-ax2c-lem1 47779

Description: First lemma for the derivation of jarr 107, imim1 84, and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7, from adh-minimp 47778 and ax-mp 5. Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minimp-jarr-imim1-ax2c-lem1	⊢ ((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃)))

Proof of Theorem adh-minimp-jarr-imim1-ax2c-lem1

Step	Hyp	Ref	Expression
1		adh-minimp 47778	. 2 ⊢ (𝜂 → ((𝜁 → 𝜎) → (((𝜌 → 𝜁) → (𝜎 → 𝜇)) → (𝜁 → 𝜇))))
2		adh-minimp 47778	. 2 ⊢ ((𝜂 → ((𝜁 → 𝜎) → (((𝜌 → 𝜁) → (𝜎 → 𝜇)) → (𝜁 → 𝜇)))) → ((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃)))

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-lem2  47780  adh-minimp-sylsimp  47782  adh-minimp-imim1  47784  adh-minimp-ax2c  47785