Theorem frege67b

Description: Lemma for frege68b . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege67b	⊢ ( ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) ) → ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege58b	⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 )
2		frege7	⊢ ( ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) → ( ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) ) → ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝑦 / 𝑥 ] 𝜑 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) ) → ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )