Theorem euex 2030

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 1507	. . 3 |- ( ph -> A. y ph )
2	1	eu1 2025	. 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 1330   E.wex 1469   [wsb 1736   E!weu 2000

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003

This theorem is referenced by:  eu2  2044  eu3h  2045  eu5  2047  exmoeudc  2063  eupickbi  2082  2eu2ex  2089  euxfrdc  2874  repizf  4052  eusvnf  4382  eusvnfb  4383  tz6.12c  5459  ndmfvg  5460  nfvres  5462  0fv  5464  eusvobj2  5768  fnoprabg  5880  txcn  12483