Theorem frege55lem1a

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege55lem1a	$$⊢\left(\tau \to if-\left(\psi ,\phi ,\neg \phi \right)\right)\to \left(\tau \to \left(\psi \leftrightarrow \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege54cor0a	$$⊢\left(\psi \leftrightarrow \phi \right)\leftrightarrow if-\left(\psi ,\phi ,\neg \phi \right)$$
2	1	biimpri	$$⊢if-\left(\psi ,\phi ,\neg \phi \right)\to \left(\psi \leftrightarrow \phi \right)$$
3	2	imim2i	$$⊢\left(\tau \to if-\left(\psi ,\phi ,\neg \phi \right)\right)\to \left(\tau \to \left(\psi \leftrightarrow \phi \right)\right)$$