Theorem eu3v

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994) Replace a nonfreeness hypothesis with a disjoint variable condition on ph , y to reduce axiom usage. (Revised by Wolf Lammen, 29-May-2019)

	Ref	Expression
Assertion	eu3v	|- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-eu	|- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		df-mo	|- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
3	2	anbi2i	|- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
4	1 3	bitri	|- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )