Theorem euen1

Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014)

	Ref	Expression
Assertion	euen1	|- ( E! x ph <-> { x | ph } ~~ 1o )

Proof

Step	Hyp	Ref	Expression
1		reuen1	|- ( E! x e. _V ph <-> { x e. _V | ph } ~~ 1o )
2		reuv	|- ( E! x e. _V ph <-> E! x ph )
3		rabab	|- { x e. _V | ph } = { x | ph }
4	3	breq1i	|- ( { x e. _V | ph } ~~ 1o <-> { x | ph } ~~ 1o )
5	1 2 4	3bitr3i	|- ( E! x ph <-> { x | ph } ~~ 1o )