Theorem rabeqf

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004)

	Ref	Expression
Hypotheses	rabeqf.1	⊢ Ⅎ 𝑥 𝐴
	rabeqf.2	⊢ Ⅎ 𝑥 𝐵
Assertion	rabeqf	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		rabeqf.1	⊢ Ⅎ 𝑥 𝐴
2		rabeqf.2	⊢ Ⅎ 𝑥 𝐵
3	1 2	nfeq	⊢ Ⅎ 𝑥 𝐴 = 𝐵
4		eleq2	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
5	4	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
6	3 5	abbid	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } )
7		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
8		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
9	6 7 8	3eqtr4g	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜑 } )