Theorem elab3

Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000) (Revised by AV, 16-Aug-2024)

	Ref	Expression
Hypotheses	elab3.1	⊢ ( 𝜓 → 𝐴 ∈ 𝑉 )
	elab3.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	elab3	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		elab3.1	⊢ ( 𝜓 → 𝐴 ∈ 𝑉 )
2		elab3.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3	2	elab3g	⊢ ( ( 𝜓 → 𝐴 ∈ 𝑉 ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )
4	1 3	ax-mp	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )