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Theorem rexdifsn 3540
Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
Assertion
Ref Expression
rexdifsn  |-  ( E. x  e.  ( A 
\  { B }
) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )

Proof of Theorem rexdifsn
StepHypRef Expression
1 eldifsn 3535 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21anbi1i 446 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( (
x  e.  A  /\  x  =/=  B )  /\  ph ) )
3 anass 393 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
42, 3bitri 182 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
54rexbii2 2382 1  |-  ( E. x  e.  ( A 
\  { B }
) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1434    =/= wne 2249   E.wrex 2354    \ cdif 2979   {csn 3416
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-in1 577  ax-in2 578  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-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-rex 2359  df-v 2612  df-dif 2984  df-sn 3422
This theorem is referenced by: (None)
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