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Theorem morex 3197
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 2720 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 exancom 1716 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( ph  /\  x  e.  A )
)
31, 2bitri 252 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( ph  /\  x  e.  A )
)
4 nfmo1 2287 . . . . . 6  |-  F/ x E* x ph
5 nfe1 1894 . . . . . 6  |-  F/ x E. x ( ph  /\  x  e.  A )
64, 5nfan 1988 . . . . 5  |-  F/ x
( E* x ph  /\ 
E. x ( ph  /\  x  e.  A ) )
7 mopick 2341 . . . . 5  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  ( ph  ->  x  e.  A ) )
86, 7alrimi 1932 . . . 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 2494 . . . . . 6  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
1210, 11imbi12d 321 . . . . 5  |-  ( x  =  B  ->  (
( ph  ->  x  e.  A )  <->  ( ps  ->  B  e.  A ) ) )
139, 12spcv 3115 . . . 4  |-  ( A. x ( ph  ->  x  e.  A )  -> 
( ps  ->  B  e.  A ) )
148, 13syl 17 . . 3  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  ( ps  ->  B  e.  A ) )
153, 14sylan2b 477 . 2  |-  ( ( E* x ph  /\  E. x  e.  A  ph )  ->  ( ps  ->  B  e.  A ) )
1615ancoms 454 1  |-  ( ( E. x  e.  A  ph 
/\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   E*wmo 2277   E.wrex 2715   _Vcvv 3022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rex 2720  df-v 3024
This theorem is referenced by: (None)
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