Theorem hbeu1 2052

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 2045	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		hba1 1551	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x A. x ( ph <-> x = y ) )
3	2	hbex 1647	. 2 |- ( E. y A. x ( ph <-> x = y ) -> A. x E. y A. x ( ph <-> x = y ) )
4	1, 3	hbxfrbi 1483	1 |- ( E! x ph -> A. x E! x ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   E.wex 1503   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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-eu 2045

This theorem is referenced by:  hbmo1  2080  eupicka  2122  exists2  2139