Theorem minimp-pm2.43 1621

Description: Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1616. It uses the classical derivation from ax-1 6 and ax-2 7 written DD22D21 in D-notation (see head comment for an explanation) and shortens the proof using mp2 9 (which only requires ax-mp 5). (Contributed by BJ, 31-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
minimp-pm2.43	⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))

Proof of Theorem minimp-pm2.43

Step	Hyp	Ref	Expression
1		minimp-ax2 1620	. 2 ⊢ ((𝜑 → (𝜑 → 𝜓)) → ((𝜑 → 𝜑) → (𝜑 → 𝜓)))
2		minimp-ax1 1618	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜑))
3		minimp-ax2 1620	. . 3 ⊢ ((𝜑 → ((𝜑 → 𝜓) → 𝜑)) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜑)))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜑))
5		minimp-ax2 1620	. 2 ⊢ (((𝜑 → (𝜑 → 𝜓)) → ((𝜑 → 𝜑) → (𝜑 → 𝜓))) → (((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜑)) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))))
6	1, 4, 5	mp2 9	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by: (None)