Theorem impsingle-step22 1665

Description: Derivation of impsingle-step22 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 22 in Lukasiewicz, where it appears as 'Cpp' 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-step22	⊢ (𝜑 → 𝜑)

Proof of Theorem impsingle-step22

Step	Hyp	Ref	Expression
1		impsingle-step4 1657	. 2 ⊢ (((𝜃 → 𝜇) → 𝜃) → (𝜆 → 𝜃))
2		impsingle-step4 1657	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜑 → 𝜑))
3		impsingle-step4 1657	. . . 4 ⊢ (((𝜑 → 𝜑) → 𝜑) → ((𝜑 → 𝜓) → 𝜑))
4		impsingle 1656	. . . 4 ⊢ ((((𝜑 → 𝜑) → 𝜑) → ((𝜑 → 𝜓) → 𝜑)) → ((((𝜑 → 𝜓) → 𝜑) → (𝜑 → 𝜑)) → ((((𝜃 → 𝜇) → 𝜃) → (𝜆 → 𝜃)) → (𝜑 → 𝜑))))
5	3, 4	ax-mp 5	. . 3 ⊢ ((((𝜑 → 𝜓) → 𝜑) → (𝜑 → 𝜑)) → ((((𝜃 → 𝜇) → 𝜃) → (𝜆 → 𝜃)) → (𝜑 → 𝜑)))
6	2, 5	ax-mp 5	. 2 ⊢ ((((𝜃 → 𝜇) → 𝜃) → (𝜆 → 𝜃)) → (𝜑 → 𝜑))
7	1, 6	ax-mp 5	1 ⊢ (𝜑 → 𝜑)

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-step25  1666  impsingle-peirce  1668