Theorem rexsngf

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012) (Revised by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	rexsngf.1	⊢ Ⅎ 𝑥 𝜓
	rexsngf.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rexsngf	⊢ ( 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		rexsns	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 )
4	1 2	sbciegf	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
5	3 4	syl5bb	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )