Theorem euex 2036

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 1506	. . 3 |- ( ph -> A. y ph )
2	1	eu1 2031	. 2 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )
3		exsimpl 1597	. 2 |- ( E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) -> E. x ph )
4	2, 3	sylbi 120	1 |- ( E! x ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1333   E.wex 1472   [wsb 1742   E!weu 2006

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009

This theorem is referenced by:  eu2  2050  eu3h  2051  eu5  2053  exmoeudc  2069  eupickbi  2088  2eu2ex  2095  euxfrdc  2898  repizf  4080  eusvnf  4411  eusvnfb  4412  tz6.12c  5495  ndmfvg  5496  nfvres  5498  0fv  5500  eusvobj2  5804  fnoprabg  5916  txcn  12635