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Theorem rexdifpr 39141
Description: Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
Assertion
Ref Expression
rexdifpr  |-  ( E. x  e.  ( A 
\  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/= 
C  /\  ph ) )

Proof of Theorem rexdifpr
StepHypRef Expression
1 eldifpr 3982 . . . . 5  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  x  =/= 
B  /\  x  =/=  C ) )
2 3anass 1011 . . . . 5  |-  ( ( x  e.  A  /\  x  =/=  B  /\  x  =/=  C )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  x  =/=  C ) ) )
31, 2bitri 257 . . . 4  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) ) )
43anbi1i 709 . . 3  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) )  /\  ph ) )
5 anass 661 . . . 4  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
) )
6 df-3an 1009 . . . . . 6  |-  ( ( x  =/=  B  /\  x  =/=  C  /\  ph ) 
<->  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)
76bicomi 207 . . . . 5  |-  ( ( ( x  =/=  B  /\  x  =/=  C
)  /\  ph )  <->  ( x  =/=  B  /\  x  =/= 
C  /\  ph ) )
87anbi2i 708 . . . 4  |-  ( ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph )
) )
95, 8bitri 257 . . 3  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
104, 9bitri 257 . 2  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
1110rexbii2 2879 1  |-  ( E. x  e.  ( A 
\  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/= 
C  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007    e. wcel 1904    =/= wne 2641   E.wrex 2757    \ cdif 3387   {cpr 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-sn 3960  df-pr 3962
This theorem is referenced by:  usgra2pth0  40177
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