Theorem rexeqf

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003) (Revised by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Hypotheses	raleq1f.1	$$⊢\underset{\_}{Ⅎ}xA$$
	raleq1f.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	rexeqf	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		raleq1f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		raleq1f.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	exbid	$$⊢A=B\to \left(\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\left(x\in B\wedge \phi \right)\right)$$
7		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
8		df-rex	$$⊢\exists x\in B\phi \leftrightarrow \exists x\left(x\in B\wedge \phi \right)$$
9	6 7 8	3bitr4g	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\phi \right)$$