Theorem frege57a

Description: Analogue of frege57aid . Proposition 57 of Frege1879 p. 51. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege57a	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜓 , 𝜒 , 𝜃 ) → if- ( 𝜑 , 𝜒 , 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege52a	⊢ ( ( 𝜓 ↔ 𝜑 ) → ( if- ( 𝜓 , 𝜒 , 𝜃 ) → if- ( 𝜑 , 𝜒 , 𝜃 ) ) )
2		frege56a	⊢ ( ( ( 𝜓 ↔ 𝜑 ) → ( if- ( 𝜓 , 𝜒 , 𝜃 ) → if- ( 𝜑 , 𝜒 , 𝜃 ) ) ) → ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜓 , 𝜒 , 𝜃 ) → if- ( 𝜑 , 𝜒 , 𝜃 ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜓 , 𝜒 , 𝜃 ) → if- ( 𝜑 , 𝜒 , 𝜃 ) ) )