Theorem adh-minimp-jarr-ax2c-lem3 44371

Description: Third lemma for the derivation of jarr 106 and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7 proper , from adh-minimp 44368 and ax-mp 5. Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minimp-jarr-ax2c-lem3	⊢ ((((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏) → 𝜏)

Proof of Theorem adh-minimp-jarr-ax2c-lem3

Step	Hyp	Ref	Expression
1		adh-minimp-jarr-lem2 44370	. 2 ⊢ (((𝜂 → ((𝜁 → 𝜎) → (((𝜌 → 𝜁) → (𝜎 → 𝜇)) → (𝜁 → 𝜇)))) → (((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏)) → (((𝜁 → 𝜎) → (((𝜌 → 𝜁) → (𝜎 → 𝜇)) → (𝜁 → 𝜇))) → 𝜏))
2		adh-minimp-jarr-lem2 44370	. 2 ⊢ ((((𝜂 → ((𝜁 → 𝜎) → (((𝜌 → 𝜁) → (𝜎 → 𝜇)) → (𝜁 → 𝜇)))) → (((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏)) → (((𝜁 → 𝜎) → (((𝜌 → 𝜁) → (𝜎 → 𝜇)) → (𝜁 → 𝜇))) → 𝜏)) → ((((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏) → 𝜏))
3	1, 2	ax-mp 5	1 ⊢ ((((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏) → 𝜏)

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-minimp-sylsimp  44372  adh-minimp-ax2c  44375