Theorem frege7

Description: A closed form of syl6 . 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.)

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

Proof

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