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Theorem eqeuOLD 26326
Description: A condition which implies existential uniqueness. (Moved into main set.mm as eqeu 2949 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
eqeuOLD.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
eqeuOLD  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem eqeuOLD
StepHypRef Expression
1 eqeuOLD.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21eqeu 2949 1  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696   E!weu 2156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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