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Theorem reean 2557
Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Hypotheses
Ref Expression
reean.1  |-  F/ y
ph
reean.2  |-  F/ x ps
Assertion
Ref Expression
reean  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps ) )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x)    B( y)

Proof of Theorem reean
StepHypRef Expression
1 an4 556 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( ph  /\ 
ps ) )  <->  ( (
x  e.  A  /\  ph )  /\  ( y  e.  B  /\  ps ) ) )
212exbii 1553 . . 3  |-  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ( ph  /\ 
ps ) )  <->  E. x E. y ( ( x  e.  A  /\  ph )  /\  ( y  e.  B  /\  ps )
) )
3 nfv 1476 . . . . 5  |-  F/ y  x  e.  A
4 reean.1 . . . . 5  |-  F/ y
ph
53, 4nfan 1512 . . . 4  |-  F/ y ( x  e.  A  /\  ph )
6 nfv 1476 . . . . 5  |-  F/ x  y  e.  B
7 reean.2 . . . . 5  |-  F/ x ps
86, 7nfan 1512 . . . 4  |-  F/ x
( y  e.  B  /\  ps )
95, 8eean 1866 . . 3  |-  ( E. x E. y ( ( x  e.  A  /\  ph )  /\  (
y  e.  B  /\  ps ) )  <->  ( E. x ( x  e.  A  /\  ph )  /\  E. y ( y  e.  B  /\  ps ) ) )
102, 9bitri 183 . 2  |-  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ( ph  /\ 
ps ) )  <->  ( E. x ( x  e.  A  /\  ph )  /\  E. y ( y  e.  B  /\  ps ) ) )
11 r2ex 2414 . 2  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ( ph  /\  ps ) ) )
12 df-rex 2381 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
13 df-rex 2381 . . 3  |-  ( E. y  e.  B  ps  <->  E. y ( y  e.  B  /\  ps )
)
1412, 13anbi12i 451 . 2  |-  ( ( E. x  e.  A  ph 
/\  E. y  e.  B  ps )  <->  ( E. x
( x  e.  A  /\  ph )  /\  E. y ( y  e.  B  /\  ps )
) )
1510, 11, 143bitr4i 211 1  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   F/wnf 1404   E.wex 1436    e. wcel 1448   E.wrex 2376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381
This theorem is referenced by:  reeanv  2558
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