Theorem hbeu1 1952

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 1945	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		hba1 1474	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x A. x ( ph <-> x = y ) )
3	2	hbex 1568	. 2 |- ( E. y A. x ( ph <-> x = y ) -> A. x E. y A. x ( ph <-> x = y ) )
4	1, 3	hbxfrbi 1402	1 |- ( E! x ph -> A. x E! x ph )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   E.wex 1422   E!weu 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-eu 1945

This theorem is referenced by:  hbmo1  1980  eupicka  2022  exists2  2039