Theorem eqabi

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 26-May-1993) Avoid ax-11 . (Revised by Wolf Lammen, 6-May-2023)

	Ref	Expression
Hypothesis	eqabi.1	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝜑 )
Assertion	eqabi	⊢ 𝐴 = { 𝑥 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		eqabi.1	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝜑 )
2	1	a1i	⊢ ( ⊤ → ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )
3	2	eqabdv	⊢ ( ⊤ → 𝐴 = { 𝑥 ∣ 𝜑 } )
4	3	mptru	⊢ 𝐴 = { 𝑥 ∣ 𝜑 }