Theorem eu3 2060

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eu3.1	|- F/ y ph

Assertion

Ref	Expression
eu3	|- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem eu3

Step	Hyp	Ref	Expression
1		eu3.1	. . 3 |- F/ y ph
2	1	nfri 1507	. 2 |- ( ph -> A. y ph )
3	2	eu3h 2059	1 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   F/wnf 1448   E.wex 1480   E!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017

This theorem is referenced by:  eqeu  2895  reu3  2915  eunex  4537