Theorem frege7 41416

Description: A closed form of syl6 35. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege7	⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓))))

Proof of Theorem frege7

Step	Hyp	Ref	Expression
1		frege5 41408	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜃 → 𝜑) → (𝜃 → 𝜓)))
2		frege6 41414	. 2 ⊢ (((𝜑 → 𝜓) → ((𝜃 → 𝜑) → (𝜃 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓))))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399

This theorem is referenced by:  frege32  41443  frege67a  41493  frege67b  41520  frege67c  41538  frege94  41565  frege107  41578  frege113  41584