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Theorem morex 3061
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 2655 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 exancom 1593 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( ph  /\  x  e.  A )
)
31, 2bitri 241 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( ph  /\  x  e.  A )
)
4 nfmo1 2249 . . . . . 6  |-  F/ x E* x ph
5 nfe1 1739 . . . . . 6  |-  F/ x E. x ( ph  /\  x  e.  A )
64, 5nfan 1836 . . . . 5  |-  F/ x
( E* x ph  /\ 
E. x ( ph  /\  x  e.  A ) )
7 mopick 2300 . . . . 5  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  ( ph  ->  x  e.  A ) )
86, 7alrimi 1773 . . . 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 2447 . . . . . 6  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
1210, 11imbi12d 312 . . . . 5  |-  ( x  =  B  ->  (
( ph  ->  x  e.  A )  <->  ( ps  ->  B  e.  A ) ) )
139, 12spcv 2985 . . . 4  |-  ( A. x ( ph  ->  x  e.  A )  -> 
( ps  ->  B  e.  A ) )
148, 13syl 16 . . 3  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  ( ps  ->  B  e.  A ) )
153, 14sylan2b 462 . 2  |-  ( ( E* x ph  /\  E. x  e.  A  ph )  ->  ( ps  ->  B  e.  A ) )
1615ancoms 440 1  |-  ( ( E. x  e.  A  ph 
/\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   E*wmo 2239   E.wrex 2650   _Vcvv 2899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-v 2901
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