Theorem impsingle-step15 1664

Description: Derivation of impsingle-step15 from ax-mp 5 and impsingle 1660. It is used as a lemma in proofs of imim1 84 and peirce 205 from impsingle 1660. It is Step 15 in Lukasiewicz, where it appears as 'CCCrqCspCCrpCsp' 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-step15	⊢ (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))

Proof of Theorem impsingle-step15

Step	Hyp	Ref	Expression
1		impsingle 1660	. 2 ⊢ (((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))
2		impsingle 1660	. . 3 ⊢ (((𝜏 → 𝜎) → 𝜌) → ((𝜌 → 𝜏) → (𝜇 → 𝜏)))
3		impsingle 1660	. . . 4 ⊢ (((((𝜑 → 𝜃) → (𝜒 → 𝜃)) → 𝜂) → ((𝜃 → 𝜆) → 𝜑)) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))))
4		impsingle 1660	. . . . . . . . 9 ⊢ ((((𝜒 → 𝜃) → 𝜁) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))
5		impsingle-step8 1662	. . . . . . . . 9 ⊢ (((((𝜒 → 𝜃) → 𝜁) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))))
6	4, 5	ax-mp 5	. . . . . . . 8 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))
7		impsingle 1660	. . . . . . . 8 ⊢ (((𝜑 → 𝜓) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → (((((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → 𝜑) → ((𝜃 → 𝜆) → 𝜑)))
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ (((((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → 𝜑) → ((𝜃 → 𝜆) → 𝜑))
9		impsingle 1660	. . . . . . 7 ⊢ ((((((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → 𝜑) → ((𝜃 → 𝜆) → 𝜑)) → ((((𝜃 → 𝜆) → 𝜑) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ ((((𝜃 → 𝜆) → 𝜑) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))))
11		impsingle 1660	. . . . . 6 ⊢ (((((𝜃 → 𝜆) → 𝜑) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))) → ((((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → ((𝜃 → 𝜆) → 𝜑)) → ((((𝜑 → 𝜃) → (𝜒 → 𝜃)) → 𝜂) → ((𝜃 → 𝜆) → 𝜑))))
12	10, 11	ax-mp 5	. . . . 5 ⊢ ((((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → ((𝜃 → 𝜆) → 𝜑)) → ((((𝜑 → 𝜃) → (𝜒 → 𝜃)) → 𝜂) → ((𝜃 → 𝜆) → 𝜑)))
13		impsingle 1660	. . . . 5 ⊢ (((((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))) → ((𝜃 → 𝜆) → 𝜑)) → ((((𝜑 → 𝜃) → (𝜒 → 𝜃)) → 𝜂) → ((𝜃 → 𝜆) → 𝜑))) → ((((((𝜑 → 𝜃) → (𝜒 → 𝜃)) → 𝜂) → ((𝜃 → 𝜆) → 𝜑)) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))) → ((((𝜏 → 𝜎) → 𝜌) → ((𝜌 → 𝜏) → (𝜇 → 𝜏))) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))))))
14	12, 13	ax-mp 5	. . . 4 ⊢ ((((((𝜑 → 𝜃) → (𝜒 → 𝜃)) → 𝜂) → ((𝜃 → 𝜆) → 𝜑)) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))) → ((((𝜏 → 𝜎) → 𝜌) → ((𝜌 → 𝜏) → (𝜇 → 𝜏))) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))))
15	3, 14	ax-mp 5	. . 3 ⊢ ((((𝜏 → 𝜎) → 𝜌) → ((𝜌 → 𝜏) → (𝜇 → 𝜏))) → ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))))
16	2, 15	ax-mp 5	. 2 ⊢ ((((𝜃 → 𝜆) → 𝜑) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) → (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))))
17	1, 16	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))

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-step18  1665  impsingle-step21  1668  impsingle-step25  1670