Theorem rabeq2i

Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004)

	Ref	Expression
Hypothesis	rabeq2i.1	⊢ 𝐴 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
Assertion	rabeq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		rabeq2i.1	⊢ 𝐴 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
2	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } )
3		rabid	⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
4	2 3	bitri	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )