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Theorem minimp-pm2.43 1562
Description: Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1557. 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 → (𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem minimp-pm2.43
StepHypRef Expression
1 minimp-ax2 1561 . 2 ((𝜑 → (𝜑𝜓)) → ((𝜑𝜑) → (𝜑𝜓)))
2 minimp-ax1 1559 . . 3 (𝜑 → ((𝜑𝜓) → 𝜑))
3 minimp-ax2 1561 . . 3 ((𝜑 → ((𝜑𝜓) → 𝜑)) → ((𝜑 → (𝜑𝜓)) → (𝜑𝜑)))
42, 3ax-mp 5 . 2 ((𝜑 → (𝜑𝜓)) → (𝜑𝜑))
5 minimp-ax2 1561 . 2 (((𝜑 → (𝜑𝜓)) → ((𝜑𝜑) → (𝜑𝜓))) → (((𝜑 → (𝜑𝜓)) → (𝜑𝜑)) → ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))))
61, 4, 5mp2 9 1 ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
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)
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