Theorem frege3

Description: Add antecedent to ax-frege2 . Special case of rp-frege3g . Proposition 3 of Frege1879 p. 29. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege3	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege2	⊢ ( ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) )
2		ax-frege1	⊢ ( ( ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) ) )