Theorem hbeu1 2009

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)

Assertion

Ref	Expression
hbeu1	|- ( E! x ph -> A. x E! x ph )

Proof of Theorem hbeu1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2002	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		hba1 1520	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x A. x ( ph <-> x = y ) )
3	2	hbex 1615	. 2 |- ( E. y A. x ( ph <-> x = y ) -> A. x E. y A. x ( ph <-> x = y ) )
4	1, 3	hbxfrbi 1448	1 |- ( E! x ph -> A. x E! x ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   E.wex 1468   E!weu 1999

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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-eu 2002

This theorem is referenced by:  hbmo1  2037  eupicka  2079  exists2  2096