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Theorem morex 3143
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 2721 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 exancom 1638 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( ph  /\  x  e.  A )
)
31, 2bitri 249 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( ph  /\  x  e.  A )
)
4 nfmo1 2267 . . . . . 6  |-  F/ x E* x ph
5 nfe1 1778 . . . . . 6  |-  F/ x E. x ( ph  /\  x  e.  A )
64, 5nfan 1861 . . . . 5  |-  F/ x
( E* x ph  /\ 
E. x ( ph  /\  x  e.  A ) )
7 mopick 2339 . . . . 5  |-  ( ( E* x ph  /\  E. x ( ph  /\  x  e.  A )
)  ->  ( ph  ->  x  e.  A ) )
86, 7alrimi 1811 . . . 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 2503 . . . . . 6  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
1210, 11imbi12d 320 . . . . 5  |-  ( x  =  B  ->  (
( ph  ->  x  e.  A )  <->  ( ps  ->  B  e.  A ) ) )
139, 12spcv 3063 . . . 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 475 . 2  |-  ( ( E* x ph  /\  E. x  e.  A  ph )  ->  ( ps  ->  B  e.  A ) )
1615ancoms 453 1  |-  ( ( E. x  e.  A  ph 
/\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E*wmo 2254   E.wrex 2716   _Vcvv 2972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-v 2974
This theorem is referenced by: (None)
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