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Metamath Proof Explorer

Theorem eqabri

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996) (Proof shortened by Wolf Lammen, 15-Nov-2019)

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

Proof

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