Theorem equsv

Description: If a formula does not contain a variable x , then it is equivalent to the corresponding prototype of substitution with a fresh variable (see sb6 ). (Contributed by BJ, 23-Jul-2023)

	Ref	Expression
Assertion	equsv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜑 ) )
2		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
3	2	a1bi	⊢ ( 𝜑 ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜑 ) )
4	1 3	bitr4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜑 )