Theorem minimp-ax2 1658

Description: Derivation of ax-2 7 from ax-mp 5 and minimp 1654. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem minimp-ax2

Step	Hyp	Ref	Expression
1		minimp-ax2c 1657	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
2		minimp-ax2c 1657	. . 3 ⊢ (((𝜑 → 𝜓) → (𝜑 → (𝜓 → 𝜒))) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
3		minimp-syllsimp 1655	. . 3 ⊢ ((((𝜑 → 𝜓) → (𝜑 → (𝜓 → 𝜒))) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
5		minimp-ax2c 1657	. . 3 ⊢ (((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))) → (((𝜑 → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))))
6		minimp-syllsimp 1655	. . 3 ⊢ ((((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))) → (((𝜑 → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → (((𝜑 → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))))
7	5, 6	ax-mp 5	. 2 ⊢ (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → (((𝜑 → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))))
8	1, 4, 7	mp2 9	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 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:  minimp-pm2.43  1659