Theorem impsingle-step21 1643

Description: Derivation of impsingle-step21 from ax-mp 5 and impsingle 1635. It is used as a lemma in the proof of imim1 83 from impsingle 1635. It is Step 21 in Lukasiewicz, where it appears as 'CCCCprqqCCqrCpr' 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-step21	⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) → ((𝜒 → 𝜓) → (𝜑 → 𝜓)))

Proof of Theorem impsingle-step21

Step	Hyp	Ref	Expression
1		impsingle-step15 1639	. 2 ⊢ (((𝜒 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) → ((𝜒 → 𝜓) → (𝜑 → 𝜓)))
2		impsingle-step20 1642	. 2 ⊢ ((((𝜒 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) → ((𝜒 → 𝜓) → (𝜑 → 𝜓))) → ((((𝜑 → 𝜓) → 𝜒) → 𝜒) → ((𝜒 → 𝜓) → (𝜑 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) → ((𝜒 → 𝜓) → (𝜑 → 𝜓)))

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:  impsingle-imim1  1646