Theorem euex 2072

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 1537	. . 3 |- ( ph -> A. y ph )
2	1	eu1 2067	. 2 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )
3		exsimpl 1628	. 2 |- ( E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) -> E. x ph )
4	2, 3	sylbi 121	1 |- ( E! x ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1503   [wsb 1773   E!weu 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045

This theorem is referenced by:  eu2  2086  eu3h  2087  eu5  2089  exmoeudc  2105  eupickbi  2124  2eu2ex  2131  euxfrdc  2947  repizf  4146  eusvnf  4485  eusvnfb  4486  tz6.12c  5585  ndmfvg  5586  elfvm  5588  nfvres  5589  0fv  5591  eusvobj2  5905  fnoprabg  6020  0g0  12962  txcn  14454