Theorem frege55cor1a

Description: Proposition 55 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege55cor1a	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		frege55a	$$⊢\left(\phi \leftrightarrow \psi \right)\to if-\left(\psi ,\phi ,\neg \phi \right)$$
2		frege55lem1a	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\to if-\left(\psi ,\phi ,\neg \phi \right)\right)\to \left(\left(\phi \leftrightarrow \psi \right)\to \left(\psi \leftrightarrow \phi \right)\right)$$
3	1 2	ax-mp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \leftrightarrow \phi \right)$$