Theorem frege5

Description: A closed form of syl . Identical to imim2 . Theorem *2.05 of WhiteheadRussell p. 100. Proposition 5 of Frege1879 p. 32. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege5	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege1	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜑 → 𝜓 ) ) )
2		frege4	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) )