Theorem euex 1973

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 1460	. . 3 |- ( ph -> A. y ph )
2	1	eu1 1968	. 2 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )
3		exsimpl 1549	. 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 1283   E.wex 1422   [wsb 1687   E!weu 1943

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946

This theorem is referenced by:  eu2  1987  eu3h  1988  eu5  1990  exmoeudc  2006  eupickbi  2025  2eu2ex  2032  euxfrdc  2787  repizf  3914  eusvnf  4231  eusvnfb  4232  tz6.12c  5255  ndmfvg  5256  nfvres  5258  0fv  5260  eusvobj2  5549  fnoprabg  5653