Theorem rabeqbidvaOLD

Description: Obsolete version of rabeqbidva as of 1-Sep-2025. (Contributed by Mario Carneiro, 26-Jan-2017) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbidvaOLD.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		rabeqbidvaOLD.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	rabbidva	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )
4	1	rabeqdv	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜒 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )
5	3 4	eqtrd	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )