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Theorem morex 2871
Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
morex.1  |-  B  e. 
_V
morex.2  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
morex  |-  ( ( 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 morex
StepHypRef Expression
1 df-rex 2423 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 exancom 1588 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( ph  /\  x  e.  A )
)
31, 2bitri 183 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( ph  /\  x  e.  A )
)
4 nfmo1 2012 . . . . . 6  |-  F/ x E* x ph
5 nfe1 1473 . . . . . 6  |-  F/ x E. x ( ph  /\  x  e.  A )
64, 5nfan 1545 . . . . 5  |-  F/ x
( E* x ph  /\ 
E. x ( ph  /\  x  e.  A ) )
7 mopick 2078 . . . . 5  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  ( ph  ->  x  e.  A ) )
86, 7alrimi 1503 . . . 4  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  A. x
( ph  ->  x  e.  A ) )
9 morex.1 . . . . 5  |-  B  e. 
_V
10 morex.2 . . . . . 6  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
11 eleq1 2203 . . . . . 6  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
1210, 11imbi12d 233 . . . . 5  |-  ( x  =  B  ->  (
( ph  ->  x  e.  A )  <->  ( ps  ->  B  e.  A ) ) )
139, 12spcv 2782 . . . 4  |-  ( A. x ( ph  ->  x  e.  A )  -> 
( ps  ->  B  e.  A ) )
148, 13syl 14 . . 3  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  ( ps  ->  B  e.  A ) )
153, 14sylan2b 285 . 2  |-  ( ( E* x ph  /\  E. x  e.  A  ph )  ->  ( ps  ->  B  e.  A ) )
1615ancoms 266 1  |-  ( ( E. x  e.  A  ph 
/\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1330    = wceq 1332   E.wex 1469    e. wcel 1481   E*wmo 2001   E.wrex 2418   _Vcvv 2689
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691
This theorem is referenced by: (None)
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