Theorem frege56a

Description: Proposition 56 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege56a	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) → if- ( 𝜓 , 𝜒 , 𝜃 ) ) ) → ( ( 𝜓 ↔ 𝜑 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) → if- ( 𝜓 , 𝜒 , 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege55cor1a	⊢ ( ( 𝜓 ↔ 𝜑 ) → ( 𝜑 ↔ 𝜓 ) )
2		frege9	⊢ ( ( ( 𝜓 ↔ 𝜑 ) → ( 𝜑 ↔ 𝜓 ) ) → ( ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) → if- ( 𝜓 , 𝜒 , 𝜃 ) ) ) → ( ( 𝜓 ↔ 𝜑 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) → if- ( 𝜓 , 𝜒 , 𝜃 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) → if- ( 𝜓 , 𝜒 , 𝜃 ) ) ) → ( ( 𝜓 ↔ 𝜑 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) → if- ( 𝜓 , 𝜒 , 𝜃 ) ) ) )