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Theorem rexdifsn 3661
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 3656 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21anbi1i 454 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( (
x  e.  A  /\  x  =/=  B )  /\  ph ) )
3 anass 399 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
42, 3bitri 183 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  /\  ph )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  ph ) ) )
54rexbii2 2449 1  |-  ( E. x  e.  ( A 
\  { B }
) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1481    =/= wne 2309   E.wrex 2418    \ cdif 3071   {csn 3530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-rex 2423  df-v 2691  df-dif 3076  df-sn 3536
This theorem is referenced by: (None)
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