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Theorem eqeu 2771
Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
Hypothesis
Ref Expression
eqeu.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
eqeu  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem eqeu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeu.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21spcegv 2695 . . . 4  |-  ( A  e.  B  ->  ( ps  ->  E. x ph )
)
32imp 122 . . 3  |-  ( ( A  e.  B  /\  ps )  ->  E. x ph )
433adant3 959 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. x ph )
5 eqeq2 2092 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
65imbi2d 228 . . . . . 6  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
76albidv 1747 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
87spcegv 2695 . . . 4  |-  ( A  e.  B  ->  ( A. x ( ph  ->  x  =  A )  ->  E. y A. x (
ph  ->  x  =  y ) ) )
98imp 122 . . 3  |-  ( ( A  e.  B  /\  A. x ( ph  ->  x  =  A ) )  ->  E. y A. x
( ph  ->  x  =  y ) )
1093adant2 958 . 2  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E. y A. x ( ph  ->  x  =  y ) )
11 nfv 1462 . . 3  |-  F/ y
ph
1211eu3 1989 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
134, 10, 12sylanbrc 408 1  |-  ( ( A  e.  B  /\  ps  /\  A. x (
ph  ->  x  =  A ) )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612
This theorem is referenced by: (None)
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