Theorem ralsngf

Description: Restricted universal quantification over a singleton. (Contributed by NM, 14-Dec-2005) (Revised by AV, 3-Apr-2023)

	Ref	Expression
Hypotheses	rexsngf.1	⊢ Ⅎ 𝑥 𝜓
	rexsngf.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ralsngf	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		rexsngf.1	⊢ Ⅎ 𝑥 𝜓
2		rexsngf.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		ralsnsg	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
4	1 2	sbciegf	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
5	3 4	bitrd	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )