Theorem frege56aid

Description: Lemma for frege57aid . Proposition 56 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege56aid	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜓 ↔ 𝜑 ) → ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege55aid	⊢ ( ( 𝜓 ↔ 𝜑 ) → ( 𝜑 ↔ 𝜓 ) )
2		frege9	⊢ ( ( ( 𝜓 ↔ 𝜑 ) → ( 𝜑 ↔ 𝜓 ) ) → ( ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜓 ↔ 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜓 ↔ 𝜑 ) → ( 𝜑 → 𝜓 ) ) )