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Theorem rexdifpr 3611
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 3610 . . . . 5  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  x  =/= 
B  /\  x  =/=  C ) )
2 3anass 977 . . . . 5  |-  ( ( x  e.  A  /\  x  =/=  B  /\  x  =/=  C )  <->  ( x  e.  A  /\  (
x  =/=  B  /\  x  =/=  C ) ) )
31, 2bitri 183 . . . 4  |-  ( x  e.  ( A  \  { B ,  C }
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) ) )
43anbi1i 455 . . 3  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C ) )  /\  ph ) )
5 anass 399 . . . 4  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
) )
6 df-3an 975 . . . . . 6  |-  ( ( x  =/=  B  /\  x  =/=  C  /\  ph ) 
<->  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)
76bicomi 131 . . . . 5  |-  ( ( ( x  =/=  B  /\  x  =/=  C
)  /\  ph )  <->  ( x  =/=  B  /\  x  =/= 
C  /\  ph ) )
87anbi2i 454 . . . 4  |-  ( ( x  e.  A  /\  ( ( x  =/= 
B  /\  x  =/=  C )  /\  ph )
)  <->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph )
) )
95, 8bitri 183 . . 3  |-  ( ( ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
104, 9bitri 183 . 2  |-  ( ( x  e.  ( A 
\  { B ,  C } )  /\  ph ) 
<->  ( x  e.  A  /\  ( x  =/=  B  /\  x  =/=  C  /\  ph ) ) )
1110rexbii2 2481 1  |-  ( E. x  e.  ( A 
\  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/= 
C  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 973    e. wcel 2141    =/= wne 2340   E.wrex 2449    \ cdif 3118   {cpr 3584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-sn 3589  df-pr 3590
This theorem is referenced by: (None)
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