Theorem eusvobj1

Description: Specify the same object in two ways when class B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010) (Proof shortened by Mario Carneiro, 19-Nov-2016)

	Ref	Expression
Hypothesis	eusvobj1.1	$$⊢B\in \mathrm{V}$$
Assertion	eusvobj1	$$⊢\exists !x\exists y\in Ax=B\to \left(\iota x|\exists y\in Ax=B\right)=\left(\iota x|\forall y\in Ax=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eusvobj1.1	$$⊢B\in \mathrm{V}$$
2		nfeu1	$$⊢Ⅎx\exists !x\exists y\in Ax=B$$
3	1	eusvobj2	$$⊢\exists !x\exists y\in Ax=B\to \left(\exists y\in Ax=B\leftrightarrow \forall y\in Ax=B\right)$$
4	2 3	alrimi	$$⊢\exists !x\exists y\in Ax=B\to \forall x\left(\exists y\in Ax=B\leftrightarrow \forall y\in Ax=B\right)$$
5		iotabi	$$⊢\forall x\left(\exists y\in Ax=B\leftrightarrow \forall y\in Ax=B\right)\to \left(\iota x|\exists y\in Ax=B\right)=\left(\iota x|\forall y\in Ax=B\right)$$
6	4 5	syl	$$⊢\exists !x\exists y\in Ax=B\to \left(\iota x|\exists y\in Ax=B\right)=\left(\iota x|\forall y\in Ax=B\right)$$