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Theorem minimp-ax2 1555
 Description: Derivation of ax-2 7 from ax-mp 5 and minimp 1551. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
minimp-ax2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Proof of Theorem minimp-ax2
StepHypRef Expression
1 minimp-ax2c 1554 . 2 ((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
2 minimp-ax2c 1554 . . 3 (((𝜑𝜓) → (𝜑 → (𝜓𝜒))) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒))))
3 minimp-sylsimp 1552 . . 3 ((((𝜑𝜓) → (𝜑 → (𝜓𝜒))) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))) → ((𝜑 → (𝜓𝜒)) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))))
42, 3ax-mp 5 . 2 ((𝜑 → (𝜓𝜒)) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒))))
5 minimp-ax2c 1554 . . 3 (((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))) → (((𝜑 → (𝜓𝜒)) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))) → ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))))
6 minimp-sylsimp 1552 . . 3 ((((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))) → (((𝜑 → (𝜓𝜒)) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))) → ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒))))) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → (((𝜑 → (𝜓𝜒)) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))) → ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒))))))
75, 6ax-mp 5 . 2 (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → (((𝜑 → (𝜓𝜒)) → (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))) → ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))))
81, 4, 7mp2 9 1 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 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:  minimp-pm2.43  1556
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