Theorem frege67c

Description: Lemma for frege68c . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege67c	⊢ ( ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) ) → ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝐴 / 𝑥 ] 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	⊢ 𝐴 ∈ 𝐵
2	1	frege58c	⊢ ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 )
3		frege7	⊢ ( ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) → ( ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) ) → ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝐴 / 𝑥 ] 𝜑 ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) ) → ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝐴 / 𝑥 ] 𝜑 ) ) )