Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  raldifsni Structured version   Unicode version

Theorem raldifsni 26734
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 3927 . . . 4  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
21imbi1i 316 . . 3  |-  ( ( x  e.  ( A 
\  { B }
)  ->  -.  ph )  <->  ( ( x  e.  A  /\  x  =/=  B
)  ->  -.  ph )
)
3 impexp 434 . . 3  |-  ( ( ( x  e.  A  /\  x  =/=  B
)  ->  -.  ph )  <->  ( x  e.  A  -> 
( x  =/=  B  ->  -.  ph ) ) )
4 df-ne 2601 . . . . . 6  |-  ( x  =/=  B  <->  -.  x  =  B )
54imbi1i 316 . . . . 5  |-  ( ( x  =/=  B  ->  -.  ph )  <->  ( -.  x  =  B  ->  -. 
ph ) )
6 con34b 284 . . . . 5  |-  ( (
ph  ->  x  =  B )  <->  ( -.  x  =  B  ->  -.  ph ) )
75, 6bitr4i 244 . . . 4  |-  ( ( x  =/=  B  ->  -.  ph )  <->  ( ph  ->  x  =  B ) )
87imbi2i 304 . . 3  |-  ( ( x  e.  A  -> 
( x  =/=  B  ->  -.  ph ) )  <-> 
( x  e.  A  ->  ( ph  ->  x  =  B ) ) )
92, 3, 83bitri 263 . 2  |-  ( ( x  e.  ( A 
\  { B }
)  ->  -.  ph )  <->  ( x  e.  A  -> 
( ph  ->  x  =  B ) ) )
109ralbii2 2733 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    \ cdif 3317   {csn 3814
This theorem is referenced by:  islindf4  27285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-sn 3820
  Copyright terms: Public domain W3C validator