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Theorem r2exf 2876
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 2805 . 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 2609 . . . . 5  |-  F/ y  x  e.  A
4319.42 1912 . . . 4  |-  ( E. y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  E. y
( y  e.  B  /\  ph ) ) )
5 anass 649 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
65exbii 1635 . . . 4  |-  ( E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  E. y ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
7 df-rex 2805 . . . . 5  |-  ( E. y  e.  B  ph  <->  E. y ( y  e.  B  /\  ph )
)
87anbi2i 694 . . . 4  |-  ( ( x  e.  A  /\  E. y  e.  B  ph ) 
<->  ( x  e.  A  /\  E. y ( y  e.  B  /\  ph ) ) )
94, 6, 83bitr4i 277 . . 3  |-  ( E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  ( x  e.  A  /\  E. y  e.  B  ph ) )
109exbii 1635 . 2  |-  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
111, 10bitr4i 252 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1587    e. wcel 1758   F/_wnfc 2602   E.wrex 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805
This theorem is referenced by:  r2ex  2878  rexcomf  2986
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