Theorem elrab3

Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006)

	Ref	Expression
Hypothesis	elrab.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	elrab3	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		elrab.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	elrab	⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )
3	2	baib	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ 𝜓 ) )