Theorem frege57aid

Description: This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri . Proposition 57 of Frege1879 p. 51. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege57aid	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		frege52aid	⊢ ( ( 𝜓 ↔ 𝜑 ) → ( 𝜓 → 𝜑 ) )
2		frege56aid	⊢ ( ( ( 𝜓 ↔ 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )