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Theorem r2exf 2741
Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
r2alf.1  |-  F/_ y A
Assertion
Ref Expression
r2exf  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem r2exf
StepHypRef Expression
1 df-rex 2711 . 2  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
2 r2alf.1 . . . . . 6  |-  F/_ y A
32nfcri 2566 . . . . 5  |-  F/ y  x  e.  A
4319.42 1902 . . . 4  |-  ( E. y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  E. y
( y  e.  B  /\  ph ) ) )
5 anass 631 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
65exbii 1592 . . . 4  |-  ( E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  E. y ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
7 df-rex 2711 . . . . 5  |-  ( E. y  e.  B  ph  <->  E. y ( y  e.  B  /\  ph )
)
87anbi2i 676 . . . 4  |-  ( ( x  e.  A  /\  E. y  e.  B  ph ) 
<->  ( x  e.  A  /\  E. y ( y  e.  B  /\  ph ) ) )
94, 6, 83bitr4i 269 . . 3  |-  ( E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  ( x  e.  A  /\  E. y  e.  B  ph ) )
109exbii 1592 . 2  |-  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
111, 10bitr4i 244 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   F/_wnfc 2559   E.wrex 2706
This theorem is referenced by:  r2ex  2743  rexcomf  2867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711
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