Theorem ralsng

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005) (Revised by Mario Carneiro, 23-Apr-2015) (Proof shortened by AV, 7-Apr-2023)

	Ref	Expression
Hypothesis	ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ralsng	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	2 1	ralsngf	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )