Theorem euex 1927

Description: Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
euex	|- (E!xph -> E.xph)

Proof of Theorem euex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1416	. . 3 |- (ph -> A.yph)
2	1	eu1 1922	. 2 |- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))
3		exsimpl 1505	. 2 |- (E.x(ph /\ A.y([y / x]ph -> x = y)) -> E.xph)
4	2, 3	sylbi 114	1 |- (E!xph -> E.xph)

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240  E.wex 1378  [wsb 1642  E!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900

This theorem is referenced by:  eu2  1941  eu3h  1942  eu5  1944  exmoeudc  1960  eupickbi  1979  2eu2ex  1986  euxfrdc  2721  repizf  3864  eusvnf  4151  eusvnfb  4152  tz6.12c  5146  ndmfvg  5147  nfvres  5149  0fv  5151  eusvobj2  5441  fnoprabg  5544