Theorem rabeq

Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003) Avoid ax-10 , ax-11 , ax-12 . (Revised by Gino Giotto, 20-Aug-2023)

	Ref	Expression
Assertion	rabeq	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2	1	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
3	2	rabbidva2	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜑 } )