Theorem rabeqbidva

Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017) Remove DV conditions. (Revised by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbidva.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		rabeqbidva.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	rabbidva	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )
4	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
5	4	anbi1d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
6	5	rabbidva2	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜒 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )
7	3 6	eqtrd	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )