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Theorem rexcom4 2642
Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
rexcom4  |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem rexcom4
StepHypRef Expression
1 rexcom 2531 . 2  |-  ( E. x  e.  A  E. y  e.  _V  ph  <->  E. y  e.  _V  E. x  e.  A  ph )
2 rexv 2637 . . 3  |-  ( E. y  e.  _V  ph  <->  E. y ph )
32rexbii 2385 . 2  |-  ( E. x  e.  A  E. y  e.  _V  ph  <->  E. x  e.  A  E. y ph )
4 rexv 2637 . 2  |-  ( E. y  e.  _V  E. x  e.  A  ph  <->  E. y E. x  e.  A  ph )
51, 3, 43bitr3i 208 1  |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1426   E.wrex 2360   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621
This theorem is referenced by:  rexcom4a  2643  reuind  2820  iuncom4  3737  dfiun2g  3762  iunn0m  3790  iunxiun  3810  iinexgm  3990  inuni  3991  iunopab  4108  xpiundi  4496  xpiundir  4497  cnvuni  4622  dmiun  4645  elres  4748  elsnres  4749  rniun  4842  imaco  4936  coiun  4940  fun11iun  5274  abrexco  5538  imaiun  5539  fliftf  5578  rexrnmpt2  5760  oprabrexex2  5901  releldm2  5955  eroveu  6381  genpassl  7081  genpassu  7082  ltexprlemopl  7158  ltexprlemopu  7160
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