Theorem impsingle-step4 1661

Description: Derivation of impsingle-step4 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 4 in Lukasiewicz, where it appears as 'CCCpqpCsp' 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-step4	⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑))

Proof of Theorem impsingle-step4

Step	Hyp	Ref	Expression
1		impsingle 1660	. 2 ⊢ (((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏)))
2		impsingle 1660	. . 3 ⊢ (((𝜑 → 𝜃) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)))
3		impsingle 1660	. . . . 5 ⊢ (((𝜑 → 𝜓) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)))
4		impsingle 1660	. . . . 5 ⊢ ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑))) → (((((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)) → (𝜑 → 𝜓)) → ((𝜑 → 𝜃) → (𝜑 → 𝜓))))
5	3, 4	ax-mp 5	. . . 4 ⊢ (((((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)) → (𝜑 → 𝜓)) → ((𝜑 → 𝜃) → (𝜑 → 𝜓)))
6		impsingle 1660	. . . 4 ⊢ ((((((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)) → (𝜑 → 𝜓)) → ((𝜑 → 𝜃) → (𝜑 → 𝜓))) → ((((𝜑 → 𝜃) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑))) → ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)))))
7	5, 6	ax-mp 5	. . 3 ⊢ ((((𝜑 → 𝜃) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑))) → ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑))))
8	2, 7	ax-mp 5	. 2 ⊢ ((((𝜏 → 𝜂) → 𝜁) → ((𝜁 → 𝜏) → (𝜎 → 𝜏))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)))
9	1, 8	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 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-step22  1669