Theorem elrab2

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006)

	Ref	Expression
Hypotheses	elrab2.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	elrab2.2	⊢ 𝐶 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
Assertion	elrab2	⊢ ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		elrab2.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		elrab2.2	⊢ 𝐶 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
3	2	eleq2i	⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } )
4	1	elrab	⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )
5	3 4	bitri	⊢ ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )