Theorem elab2g

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995)

	Ref	Expression
Hypotheses	elab2g.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	elab2g.2	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
Assertion	elab2g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		elab2g.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		elab2g.2	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
3	2	eleq2i	⊢ ( 𝐴 ∈ 𝐵 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } )
4	1	elabg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )
5	3 4	bitrid	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ 𝜓 ) )