Theorem impsingle-step25 1645

Description: Derivation of impsingle-step25 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 25 in Lukasiewicz, where it appears as 'CCpqCCCprqq' 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-step25	⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))

Proof of Theorem impsingle-step25

Step	Hyp	Ref	Expression
1		impsingle-step22 1644	. . . 4 ⊢ ((((𝜑 → 𝜒) → 𝜓) → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))
2		impsingle-step20 1642	. . . 4 ⊢ (((((𝜑 → 𝜒) → 𝜓) → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → (((𝜓 → 𝜃) → (𝜑 → 𝜒)) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)))
3	1, 2	ax-mp 5	. . 3 ⊢ (((𝜓 → 𝜃) → (𝜑 → 𝜒)) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))
4		impsingle-step8 1637	. . 3 ⊢ ((((𝜓 → 𝜃) → (𝜑 → 𝜒)) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → ((𝜑 → 𝜒) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)))
5	3, 4	ax-mp 5	. 2 ⊢ ((𝜑 → 𝜒) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))
6		impsingle-step15 1639	. 2 ⊢ (((𝜑 → 𝜒) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)))
7	5, 6	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 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  impsingle-peirce  1647