Theorem adh-minim-ax1-ax2-lem3 43315

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

Assertion

Ref	Expression
adh-minim-ax1-ax2-lem3	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜃 → (𝜑 → 𝜒))))

Proof of Theorem adh-minim-ax1-ax2-lem3

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem1 43313	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → ((𝜃 → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → (𝜃 → (𝜑 → 𝜒)))) → (𝜓 → (𝜃 → (𝜑 → 𝜒)))))
2		adh-minim-ax1-ax2-lem2 43314	. 2 ⊢ (((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → ((𝜃 → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → (𝜃 → (𝜑 → 𝜒)))) → (𝜓 → (𝜃 → (𝜑 → 𝜒))))) → ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜃 → (𝜑 → 𝜒)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜃 → (𝜑 → 𝜒))))

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  43321