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Theorem rexdifpr 30267
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 3997 . . . . 5  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  x  =/= 
B  /\  x  =/=  C ) )
2 3anass 969 . . . . 5  |-  ( ( x  e.  A  /\  x  =/=  B  /\  x  =/=  C )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  x  =/=  C ) ) )
31, 2bitri 249 . . . 4  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) ) )
43anbi1i 695 . . 3  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) )  /\  ph ) )
5 anass 649 . . . 4  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
) )
6 df-3an 967 . . . . . 6  |-  ( ( x  =/=  B  /\  x  =/=  C  /\  ph ) 
<->  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)
76bicomi 202 . . . . 5  |-  ( ( ( x  =/=  B  /\  x  =/=  C
)  /\  ph )  <->  ( x  =/=  B  /\  x  =/= 
C  /\  ph ) )
87anbi2i 694 . . . 4  |-  ( ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph )
) )
95, 8bitri 249 . . 3  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
104, 9bitri 249 . 2  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
1110rexbii2 2860 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 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2645   E.wrex 2797    \ cdif 3428   {cpr 3982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-rex 2802  df-v 3074  df-dif 3434  df-un 3436  df-sn 3981  df-pr 3983
This theorem is referenced by:  usgra2pth0  30445
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