Theorem impsingle-step8 1662

Description: Derivation of impsingle-step8 from ax-mp 5 and impsingle 1660. It is used as a lemma in proofs of ax-1 6 imim1 84 and peirce 205 from impsingle 1660. It is Step 8 in Lukasiewicz, where it appears as 'CCCsqpCqp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
impsingle-step8	⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

Proof of Theorem impsingle-step8

Step	Hyp	Ref	Expression
1		impsingle 1660	. 2 ⊢ (((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏)))
2		impsingle 1660	. . 3 ⊢ (((𝜒 → 𝜃) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))
3		impsingle 1660	. . . . . . . 8 ⊢ (((𝜓 → 𝜃) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)))
4		impsingle 1660	. . . . . . . . . 10 ⊢ (((𝜓 → 𝜒) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)))
5		impsingle 1660	. . . . . . . . . 10 ⊢ ((((𝜓 → 𝜒) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓))) → (((((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)) → (𝜓 → 𝜒)) → ((𝜓 → 𝜃) → (𝜓 → 𝜒))))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ (((((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)) → (𝜓 → 𝜒)) → ((𝜓 → 𝜃) → (𝜓 → 𝜒)))
7		impsingle 1660	. . . . . . . . 9 ⊢ ((((((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)) → (𝜓 → 𝜒)) → ((𝜓 → 𝜃) → (𝜓 → 𝜒))) → ((((𝜓 → 𝜃) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓))) → ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)))))
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ ((((𝜓 → 𝜃) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓))) → ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓))))
9	3, 8	ax-mp 5	. . . . . . 7 ⊢ ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)))
10	1, 9	ax-mp 5	. . . . . 6 ⊢ (((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓))
11		impsingle 1660	. . . . . 6 ⊢ ((((𝜓 → 𝜒) → 𝜓) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))))
12	10, 11	ax-mp 5	. . . . 5 ⊢ (((𝜑 → 𝜓) → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))
13		impsingle 1660	. . . . 5 ⊢ ((((𝜑 → 𝜓) → (𝜓 → 𝜒)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))) → (((((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜒 → 𝜃) → (𝜑 → 𝜓))))
14	12, 13	ax-mp 5	. . . 4 ⊢ (((((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜒 → 𝜃) → (𝜑 → 𝜓)))
15		impsingle 1660	. . . 4 ⊢ ((((((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜒 → 𝜃) → (𝜑 → 𝜓))) → ((((𝜒 → 𝜃) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))) → ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))))
16	14, 15	ax-mp 5	. . 3 ⊢ ((((𝜒 → 𝜃) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))) → ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))))
17	2, 16	ax-mp 5	. 2 ⊢ ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))
18	1, 17	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  impsingle-ax1  1663  impsingle-step15  1664  impsingle-step18  1665  impsingle-step20  1667  impsingle-step25  1670