Theorem impsingle-step19 1666

Description: Derivation of impsingle-step19 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 19 in Lukasiewicz, where it appears as 'CCCCspqCrpCCCpqrCsp' 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-step19	⊢ ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓)))

Proof of Theorem impsingle-step19

Step	Hyp	Ref	Expression
1		impsingle-step18 1665	. 2 ⊢ ((((𝜏 → 𝜂) → (𝜁 → 𝜂)) → (((𝜂 → 𝜎) → 𝜏) → 𝜌)) → (𝜇 → (((𝜂 → 𝜎) → 𝜏) → 𝜌)))
2		impsingle-step18 1665	. . 3 ⊢ ((((𝜃 → 𝜓) → (𝜑 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓))) → ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓))))
3		impsingle-step18 1665	. . 3 ⊢ (((((𝜃 → 𝜓) → (𝜑 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓))) → ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓)))) → (((((𝜏 → 𝜂) → (𝜁 → 𝜂)) → (((𝜂 → 𝜎) → 𝜏) → 𝜌)) → (𝜇 → (((𝜂 → 𝜎) → 𝜏) → 𝜌))) → ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓)))))
4	2, 3	ax-mp 5	. 2 ⊢ (((((𝜏 → 𝜂) → (𝜁 → 𝜂)) → (((𝜂 → 𝜎) → 𝜏) → 𝜌)) → (𝜇 → (((𝜂 → 𝜎) → 𝜏) → 𝜌))) → ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓))))
5	1, 4	ax-mp 5	1 ⊢ ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓)))

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-step20  1667