Theorem hbeu1 2024

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 2017	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		hba1 1528	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x A. x ( ph <-> x = y ) )
3	2	hbex 1624	. 2 |- ( E. y A. x ( ph <-> x = y ) -> A. x E. y A. x ( ph <-> x = y ) )
4	1, 3	hbxfrbi 1460	1 |- ( E! x ph -> A. x E! x ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   E.wex 1480   E!weu 2014

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-eu 2017

This theorem is referenced by:  hbmo1  2052  eupicka  2094  exists2  2111