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Theorem rexdifsn 4145
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 4141 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21anbi1i 693 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( (
x  e.  A  /\  x  =/=  B )  /\  ph ) )
3 anass 647 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
42, 3bitri 249 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
54rexbii2 2954 1  |-  ( E. x  e.  ( A 
\  { B }
) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    e. wcel 1823    =/= wne 2649   E.wrex 2805    \ cdif 3458   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-v 3108  df-dif 3464  df-sn 4017
This theorem is referenced by:  symgfix2  16640  2spot2iun2spont  25093  usgra2pth0  32727  dihatexv  37462  lcfl8b  37628
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