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Theorem reean 2706
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 797 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( ph  /\ 
ps ) )  <->  ( (
x  e.  A  /\  ph )  /\  ( y  e.  B  /\  ps ) ) )
212exbii 1570 . . 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 1605 . . . . 5  |-  F/ y  x  e.  A
4 reean.1 . . . . 5  |-  F/ y
ph
53, 4nfan 1771 . . . 4  |-  F/ y ( x  e.  A  /\  ph )
6 nfv 1605 . . . . 5  |-  F/ x  y  e.  B
7 reean.2 . . . . 5  |-  F/ x ps
86, 7nfan 1771 . . . 4  |-  F/ x
( y  e.  B  /\  ps )
95, 8eean 1853 . . 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 240 . 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 2581 . 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 2549 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
13 df-rex 2549 . . 3  |-  ( E. y  e.  B  ps  <->  E. y ( y  e.  B  /\  ps )
)
1412, 13anbi12i 678 . 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 268 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:    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531    e. wcel 1684   E.wrex 2544
This theorem is referenced by:  reeanv  2707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549
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