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Theorem reu6i 2928
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
reu6i  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reu6i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2187 . . . . 5  |-  ( y  =  B  ->  (
x  =  y  <->  x  =  B ) )
21bibi2d 232 . . . 4  |-  ( y  =  B  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  B ) ) )
32ralbidv 2477 . . 3  |-  ( y  =  B  ->  ( A. x  e.  A  ( ph  <->  x  =  y
)  <->  A. x  e.  A  ( ph  <->  x  =  B
) ) )
43rspcev 2841 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E. y  e.  A  A. x  e.  A  ( ph  <->  x  =  y ) )
5 reu6 2926 . 2  |-  ( E! x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  ( ph 
<->  x  =  y ) )
64, 5sylibr 134 1  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739
This theorem is referenced by:  eqreu  2929  riota5f  5849  negeu  8135  creur  8902  creui  8903
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