ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reu6i Unicode version

Theorem reu6i 2784
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 2091 . . . . 5  |-  ( y  =  B  ->  (
x  =  y  <->  x  =  B ) )
21bibi2d 230 . . . 4  |-  ( y  =  B  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  B ) ) )
32ralbidv 2369 . . 3  |-  ( y  =  B  ->  ( A. x  e.  A  ( ph  <->  x  =  y
)  <->  A. x  e.  A  ( ph  <->  x  =  B
) ) )
43rspcev 2702 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E. y  e.  A  A. x  e.  A  ( ph  <->  x  =  y ) )
5 reu6 2782 . 2  |-  ( E! x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  ( ph 
<->  x  =  y ) )
64, 5sylibr 132 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   E!wreu 2351
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-v 2604
This theorem is referenced by:  eqreu  2785  riota5f  5517  negeu  7355  creur  8092  creui  8093
  Copyright terms: Public domain W3C validator