Theorem frege68c

Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege68c	⊢ ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝐴 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	⊢ 𝐴 ∈ 𝐵
2		frege57aid	⊢ ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) )
3	1	frege67c	⊢ ( ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜑 ) ) → ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝐴 / 𝑥 ] 𝜑 ) ) )
4	2 3	ax-mp	⊢ ( ( ∀ 𝑥 𝜑 ↔ 𝜓 ) → ( 𝜓 → [ 𝐴 / 𝑥 ] 𝜑 ) )