Theorem frege55lem1a

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege55lem1a	⊢ ( ( 𝜏 → if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) ) → ( 𝜏 → ( 𝜓 ↔ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege54cor0a	⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) )
2	1	biimpri	⊢ ( if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) → ( 𝜓 ↔ 𝜑 ) )
3	2	imim2i	⊢ ( ( 𝜏 → if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) ) → ( 𝜏 → ( 𝜓 ↔ 𝜑 ) ) )