MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r2exf Unicode version

Theorem r2exf 2592
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 2562 . 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 2426 . . . . 5  |-  F/ y  x  e.  A
4319.42 1828 . . . 4  |-  ( E. y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  E. y
( y  e.  B  /\  ph ) ) )
5 anass 630 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
65exbii 1572 . . . 4  |-  ( E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  E. y ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
7 df-rex 2562 . . . . 5  |-  ( E. y  e.  B  ph  <->  E. y ( y  e.  B  /\  ph )
)
87anbi2i 675 . . . 4  |-  ( ( x  e.  A  /\  E. y  e.  B  ph ) 
<->  ( x  e.  A  /\  E. y ( y  e.  B  /\  ph ) ) )
94, 6, 83bitr4i 268 . . 3  |-  ( E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  ( x  e.  A  /\  E. y  e.  B  ph ) )
109exbii 1572 . 2  |-  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
111, 10bitr4i 243 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 176    /\ wa 358   E.wex 1531    e. wcel 1696   F/_wnfc 2419   E.wrex 2557
This theorem is referenced by:  r2ex  2594  rexcomf  2712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562
  Copyright terms: Public domain W3C validator