Theorem frege53a

Description: Lemma for frege55a . Proposition 53 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege53a	$$⊢if-\left(\phi ,\theta ,\chi \right)\to \left(\left(\phi \leftrightarrow \psi \right)\to if-\left(\psi ,\theta ,\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-frege52a	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(if-\left(\phi ,\theta ,\chi \right)\to if-\left(\psi ,\theta ,\chi \right)\right)$$
2		ax-frege8	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\to \left(if-\left(\phi ,\theta ,\chi \right)\to if-\left(\psi ,\theta ,\chi \right)\right)\right)\to \left(if-\left(\phi ,\theta ,\chi \right)\to \left(\left(\phi \leftrightarrow \psi \right)\to if-\left(\psi ,\theta ,\chi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢if-\left(\phi ,\theta ,\chi \right)\to \left(\left(\phi \leftrightarrow \psi \right)\to if-\left(\psi ,\theta ,\chi \right)\right)$$