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- ( 𝜑 , 𝜃 , 𝜒 ) → ( ( 𝜑 ↔ 𝜓 ) → if- ( 𝜓 , 𝜃 , 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege52a	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜑 , 𝜃 , 𝜒 ) → if- ( 𝜓 , 𝜃 , 𝜒 ) ) )
2		ax-frege8	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜑 , 𝜃 , 𝜒 ) → if- ( 𝜓 , 𝜃 , 𝜒 ) ) ) → ( if- ( 𝜑 , 𝜃 , 𝜒 ) → ( ( 𝜑 ↔ 𝜓 ) → if- ( 𝜓 , 𝜃 , 𝜒 ) ) ) )
3	1 2	ax-mp	⊢ ( if- ( 𝜑 , 𝜃 , 𝜒 ) → ( ( 𝜑 ↔ 𝜓 ) → if- ( 𝜓 , 𝜃 , 𝜒 ) ) )