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	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		frege52aid	$$⊢\left(\psi \leftrightarrow \phi \right)\to \left(\psi \to \phi \right)$$
2		frege56aid	$$⊢\left(\left(\psi \leftrightarrow \phi \right)\to \left(\psi \to \phi \right)\right)\to \left(\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)\right)$$
3	1 2	ax-mp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$