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	$$⊢\underset{\_}{Ⅎ}xA$$
	rabeqf.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	rabeqf	$$⊢A=B\to \left\{x\in A|\phi \right\}=\left\{x\in B|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabeqf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		rabeqf.2	$$⊢\underset{\_}{Ⅎ}xB$$
3	1 2	nfeq	$$⊢ℲxA=B$$
4		eleq2	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$
5	4	anbi1d	$$⊢A=B\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \phi \right)\right)$$
6	3 5	abbid	$$⊢A=B\to \left\{x|\left(x\in A\wedge \phi \right)\right\}=\left\{x|\left(x\in B\wedge \phi \right)\right\}$$
7		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
8		df-rab	$$⊢\left\{x\in B|\phi \right\}=\left\{x|\left(x\in B\wedge \phi \right)\right\}$$
9	6 7 8	3eqtr4g	$$⊢A=B\to \left\{x\in A|\phi \right\}=\left\{x\in B|\phi \right\}$$