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Theorem raldifsni 26753
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 3749 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21imbi1i 315 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  ->  -.  ph )  <->  ( ( x  e.  A  /\  x  =/=  B
)  ->  -.  ph )
)
3 impexp 433 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  ->  -.  ph )  <->  ( x  e.  A  -> 
( x  =/=  B  ->  -.  ph ) ) )
4 df-ne 2448 . . . . . 6  |-  ( x  =/=  B  <->  -.  x  =  B )
54imbi1i 315 . . . . 5  |-  ( ( x  =/=  B  ->  -.  ph )  <->  ( -.  x  =  B  ->  -. 
ph ) )
6 con34b 283 . . . . 5  |-  ( (
ph  ->  x  =  B )  <->  ( -.  x  =  B  ->  -.  ph ) )
75, 6bitr4i 243 . . . 4  |-  ( ( x  =/=  B  ->  -.  ph )  <->  ( ph  ->  x  =  B ) )
87imbi2i 303 . . 3  |-  ( ( x  e.  A  -> 
( x  =/=  B  ->  -.  ph ) )  <-> 
( x  e.  A  ->  ( ph  ->  x  =  B ) ) )
92, 3, 83bitri 262 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  ->  -.  ph )  <->  ( x  e.  A  -> 
( ph  ->  x  =  B ) ) )
109ralbii2 2571 1  |-  ( A. x  e.  ( A  \  { B } )  -.  ph  <->  A. x  e.  A  ( ph  ->  x  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149   {csn 3640
This theorem is referenced by:  islindf4  27308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-sn 3646
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