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Theorem rexdifsn 4092
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 4088 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21anbi1i 709 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( (
x  e.  A  /\  x  =/=  B )  /\  ph ) )
3 anass 661 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
42, 3bitri 257 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
54rexbii2 2879 1  |-  ( E. x  e.  ( A 
\  { B }
) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    e. wcel 1904    =/= wne 2641   E.wrex 2757    \ cdif 3387   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rex 2762  df-v 3033  df-dif 3393  df-sn 3960
This theorem is referenced by:  symgfix2  17135  2spot2iun2spont  25698  dihatexv  34977  lcfl8b  35143  usgra2pth0  40177
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