Theorem elab4g

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012)

	Ref	Expression
Hypotheses	elab4g.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	elab4g.2	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
Assertion	elab4g	⊢ ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		elab4g.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		elab4g.2	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
3		elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )
4	1 2	elab2g	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝐵 ↔ 𝜓 ) )
5	3 4	biadanii	⊢ ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝜓 ) )