Theorem euex 1971

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

Assertion

Ref	Expression
euex	|- ( E! x ph -> E. x ph )

Proof of Theorem euex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1459	. . 3 |- ( ph -> A. y ph )
2	1	eu1 1966	. 2 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )
3		exsimpl 1548	. 2 |- ( E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) -> E. x ph )
4	2, 3	sylbi 119	1 |- ( E! x ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421   [wsb 1685   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944

This theorem is referenced by:  eu2  1985  eu3h  1986  eu5  1988  exmoeudc  2004  eupickbi  2023  2eu2ex  2030  euxfrdc  2778  repizf  3894  eusvnf  4203  eusvnfb  4204  tz6.12c  5224  ndmfvg  5225  nfvres  5227  0fv  5229  eusvobj2  5518  fnoprabg  5622