Theorem elab2

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

	Ref	Expression
Hypotheses	elab2.1	⊢ 𝐴 ∈ V
	elab2.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	elab2.3	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
Assertion	elab2	⊢ ( 𝐴 ∈ 𝐵 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		elab2.1	⊢ 𝐴 ∈ V
2		elab2.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		elab2.3	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
4	2 3	elab2g	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝐵 ↔ 𝜓 ) )
5	1 4	ax-mp	⊢ ( 𝐴 ∈ 𝐵 ↔ 𝜓 )