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Theorem rexdifpr 39004
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 3992 . . . . 5  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  x  =/= 
B  /\  x  =/=  C ) )
2 3anass 990 . . . . 5  |-  ( ( x  e.  A  /\  x  =/=  B  /\  x  =/=  C )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  x  =/=  C ) ) )
31, 2bitri 253 . . . 4  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) ) )
43anbi1i 702 . . 3  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) )  /\  ph ) )
5 anass 655 . . . 4  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
) )
6 df-3an 988 . . . . . 6  |-  ( ( x  =/=  B  /\  x  =/=  C  /\  ph ) 
<->  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)
76bicomi 206 . . . . 5  |-  ( ( ( x  =/=  B  /\  x  =/=  C
)  /\  ph )  <->  ( x  =/=  B  /\  x  =/= 
C  /\  ph ) )
87anbi2i 701 . . . 4  |-  ( ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph )
) )
95, 8bitri 253 . . 3  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
104, 9bitri 253 . 2  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
1110rexbii2 2889 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 188    /\ wa 371    /\ w3a 986    e. wcel 1889    =/= wne 2624   E.wrex 2740    \ cdif 3403   {cpr 3972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-rex 2745  df-v 3049  df-dif 3409  df-un 3411  df-sn 3971  df-pr 3973
This theorem is referenced by:  usgra2pth0  39773
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