Theorem hbeu1 2096

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 2089	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		hba1 1593	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x A. x ( ph <-> x = y ) )
3	2	hbex 1689	. 2 |- ( E. y A. x ( ph <-> x = y ) -> A. x E. y A. x ( ph <-> x = y ) )
4	1, 3	hbxfrbi 1525	1 |- ( E! x ph -> A. x E! x ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   E.wex 1545   E!weu 2086

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-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-eu 2089

This theorem is referenced by:  hbmo1  2124  eupicka  2167  exists2  2184