Theorem rabeqsn

Description: Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019) (Proof shortened by AV, 26-Aug-2022)

	Ref	Expression
Assertion	rabeqsn	⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } = { 𝑋 } ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) ↔ 𝑥 = 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		df-rab	⊢ { 𝑥 ∈ 𝑉 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) }
2	1	eqeq1i	⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } = { 𝑋 } ↔ { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) } = { 𝑋 } )
3		absn	⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) } = { 𝑋 } ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) ↔ 𝑥 = 𝑋 ) )
4	2 3	bitri	⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } = { 𝑋 } ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) ↔ 𝑥 = 𝑋 ) )