Theorem rabeqbii

Description: Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbii.1	⊢ 𝐴 = 𝐵
2		rabeqbii.2	⊢ ( 𝜑 ↔ 𝜓 )
3	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
4	3 2	anbi12i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
5	4	abbii	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) }
6		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
7		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) }
8	5 6 7	3eqtr4i	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 }