Theorem impsingle-step18 1661

Description: Derivation of impsingle-step18 from ax-mp 5 and impsingle 1656. It is used as a lemma in proofs of imim1 84 and peirce 205 from impsingle 1656. It is Step 18 in Lukasiewicz, where it appears as 'CCCCrpCspCCCpqrtCuCCCpqrt' 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-step18	⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → (((𝜓 → 𝜃) → 𝜑) → 𝜏)) → (𝜂 → (((𝜓 → 𝜃) → 𝜑) → 𝜏)))

Proof of Theorem impsingle-step18

Step	Hyp	Ref	Expression
1		impsingle 1656	. . 3 ⊢ (((𝜓 → 𝜃) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))
2		impsingle 1656	. . . . 5 ⊢ ((((𝜒 → 𝜓) → 𝜌) → (((𝜓 → 𝜃) → 𝜑) → 𝜏)) → (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))))
3		impsingle-step8 1658	. . . . 5 ⊢ (((((𝜒 → 𝜓) → 𝜌) → (((𝜓 → 𝜃) → 𝜑) → 𝜏)) → (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))) → ((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))))
5		impsingle-step15 1660	. . . 4 ⊢ (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))) → ((((𝜓 → 𝜃) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))) → (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))))
6	4, 5	ax-mp 5	. . 3 ⊢ ((((𝜓 → 𝜃) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))) → (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))))
7	1, 6	ax-mp 5	. 2 ⊢ (((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))
8		impsingle 1656	. 2 ⊢ ((((((𝜓 → 𝜃) → 𝜑) → 𝜏) → (𝜒 → 𝜓)) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))) → ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → (((𝜓 → 𝜃) → 𝜑) → 𝜏)) → (𝜂 → (((𝜓 → 𝜃) → 𝜑) → 𝜏))))
9	7, 8	ax-mp 5	1 ⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → (((𝜓 → 𝜃) → 𝜑) → 𝜏)) → (𝜂 → (((𝜓 → 𝜃) → 𝜑) → 𝜏)))

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-step19  1662