Theorem impsingle-imim1 1646

Description: Derivation of impsingle-imim1 (imim1 83) from ax-mp 5 and impsingle 1635. It is step 29 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
impsingle-imim1	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Proof of Theorem impsingle-imim1

Step	Hyp	Ref	Expression
1		impsingle-step21 1643	. 2 ⊢ ((((𝜑 → 𝜒) → 𝜓) → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
2		impsingle-step25 1645	. . . 4 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))
3		impsingle-step25 1645	. . . 4 ⊢ (((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → ((((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)))
4	2, 3	ax-mp 5	. . 3 ⊢ ((((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))
5		impsingle-step21 1643	. . 3 ⊢ (((((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → (((((𝜑 → 𝜒) → 𝜓) → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))))
6	4, 5	ax-mp 5	. 2 ⊢ (((((𝜑 → 𝜒) → 𝜓) → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))))
7	1, 6	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)