Theorem adh-minimp-ax2-lem4 48013

Description: Fourth lemma for the derivation of ax-2 7 from adh-minimp 48005 and ax-mp 5. Polish prefix notation: CpCCqCprCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minimp-ax2-lem4	⊢ (𝜑 → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒)))

Proof of Theorem adh-minimp-ax2-lem4

Step	Hyp	Ref	Expression
1		adh-minimp-ax2c 48012	. 2 ⊢ ((𝜓 → 𝜑) → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒)))
2		adh-minimp-sylsimp 48009	. 2 ⊢ (((𝜓 → 𝜑) → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒))) → (𝜑 → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒))))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒)))

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-ax2  48014