Theorem reueq1f

Description: Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004) (Revised by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Hypotheses	rmoeq1f.1	$$⊢\underset{\_}{Ⅎ}xA$$
	rmoeq1f.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	reueq1f	$$⊢A=B\to \left(\exists !x\in A\phi \leftrightarrow \exists !x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rmoeq1f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		rmoeq1f.2	$$⊢\underset{\_}{Ⅎ}xB$$
3	1 2	rexeqf	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\phi \right)$$
4	1 2	rmoeq1f	$$⊢A=B\to \left({\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\in B\phi \right)$$
5	3 4	anbi12d	$$⊢A=B\to \left(\left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)\leftrightarrow \left(\exists x\in B\phi \wedge {\exists }^{*}x\in B\phi \right)\right)$$
6		reu5	$$⊢\exists !x\in A\phi \leftrightarrow \left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)$$
7		reu5	$$⊢\exists !x\in B\phi \leftrightarrow \left(\exists x\in B\phi \wedge {\exists }^{*}x\in B\phi \right)$$
8	5 6 7	3bitr4g	$$⊢A=B\to \left(\exists !x\in A\phi \leftrightarrow \exists !x\in B\phi \right)$$