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Theorem raldifsni 4010
Description: Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Assertion
Ref Expression
raldifsni  |-  ( A. x  e.  ( A  \  { B } )  -.  ph  <->  A. x  e.  A  ( ph  ->  x  =  B ) )

Proof of Theorem raldifsni
StepHypRef Expression
1 eldifsn 4005 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21imbi1i 325 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  ->  -.  ph )  <->  ( ( x  e.  A  /\  x  =/=  B
)  ->  -.  ph )
)
3 impexp 446 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  ->  -.  ph )  <->  ( x  e.  A  -> 
( x  =/=  B  ->  -.  ph ) ) )
4 df-ne 2613 . . . . . 6  |-  ( x  =/=  B  <->  -.  x  =  B )
54imbi1i 325 . . . . 5  |-  ( ( x  =/=  B  ->  -.  ph )  <->  ( -.  x  =  B  ->  -. 
ph ) )
6 con34b 292 . . . . 5  |-  ( (
ph  ->  x  =  B )  <->  ( -.  x  =  B  ->  -.  ph ) )
75, 6bitr4i 252 . . . 4  |-  ( ( x  =/=  B  ->  -.  ph )  <->  ( ph  ->  x  =  B ) )
87imbi2i 312 . . 3  |-  ( ( x  e.  A  -> 
( x  =/=  B  ->  -.  ph ) )  <-> 
( x  e.  A  ->  ( ph  ->  x  =  B ) ) )
92, 3, 83bitri 271 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  ->  -.  ph )  <->  ( x  e.  A  -> 
( ph  ->  x  =  B ) ) )
109ralbii2 2748 1  |-  ( A. x  e.  ( A  \  { B } )  -.  ph  <->  A. x  e.  A  ( ph  ->  x  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720    \ cdif 3330   {csn 3882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-v 2979  df-dif 3336  df-sn 3883
This theorem is referenced by:  islindf4  18272  snlindsntor  31010
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