Theorem rabeqi

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf . (Contributed by Glauco Siliprandi, 26-Jun-2021) Avoid ax-10 , ax-11 , ax-12 . (Revised by Gino Giotto, 3-Jun-2024)

	Ref	Expression
Hypothesis	rabeqi.1	⊢ 𝐴 = 𝐵
Assertion	rabeqi	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		rabeqi.1	⊢ 𝐴 = 𝐵
2	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
3	2	anbi1i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
4	3	rabbia2	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜑 }