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Theorem reean 2969
Description: Rearrange restricted 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 838 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( ph  /\ 
ps ) )  <->  ( (
x  e.  A  /\  ph )  /\  ( y  e.  B  /\  ps ) ) )
212exbii 1730 . . 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 1772 . . . . 5  |-  F/ y  x  e.  A
4 reean.1 . . . . 5  |-  F/ y
ph
53, 4nfan 2022 . . . 4  |-  F/ y ( x  e.  A  /\  ph )
6 nfv 1772 . . . . 5  |-  F/ x  y  e.  B
7 reean.2 . . . . 5  |-  F/ x ps
86, 7nfan 2022 . . . 4  |-  F/ x
( y  e.  B  /\  ps )
95, 8eean 2088 . . 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 257 . 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 2925 . 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 2755 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
13 df-rex 2755 . . 3  |-  ( E. y  e.  B  ps  <->  E. y ( y  e.  B  /\  ps )
)
1412, 13anbi12i 708 . 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 285 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 setvar class
Syntax hints:    <-> wb 189    /\ wa 375   E.wex 1674   F/wnf 1678    e. wcel 1898   E.wrex 2750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679  df-ral 2754  df-rex 2755
This theorem is referenced by:  reeanv  2970  disjrnmpt2  37501
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