Theorem rabeqbidv

Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009)

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

Proof

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