Theorem frege55a

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

	Ref	Expression
Assertion	frege55a	⊢ ( ( 𝜑 ↔ 𝜓 ) → if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		frege54cor1a	⊢ if- ( 𝜑 , 𝜑 , ¬ 𝜑 )
2		frege53a	⊢ ( if- ( 𝜑 , 𝜑 , ¬ 𝜑 ) → ( ( 𝜑 ↔ 𝜓 ) → if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 ↔ 𝜓 ) → if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) )