Theorem euex 2068

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 2063	. 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 2038

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 2041

This theorem is referenced by:  eu2  2082  eu3h  2083  eu5  2085  exmoeudc  2101  eupickbi  2120  2eu2ex  2127  euxfrdc  2938  repizf  4134  eusvnf  4471  eusvnfb  4472  tz6.12c  5564  ndmfvg  5565  elfvm  5567  nfvres  5568  0fv  5570  eusvobj2  5883  fnoprabg  5998  0g0  12855  txcn  14252