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Theorem morexOLD 26450
Description: Derive membership from uniqueness. (Moved to morex 2962 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
morexOLD.1  |-  B  e. 
_V
morexOLD.2  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
morexOLD  |-  ( ( E. x  e.  A  ph 
/\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
Distinct variable groups:    x, B    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem morexOLD
StepHypRef Expression
1 morexOLD.1 . 2  |-  B  e. 
_V
2 morexOLD.2 . 2  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
31, 2morex 2962 1  |-  ( ( E. x  e.  A  ph 
/\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E*wmo 2157   E.wrex 2557   _Vcvv 2801
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803
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