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Theorem ralcom4 2918
Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
ralcom4  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem ralcom4
StepHypRef Expression
1 ralcom 2812 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. y  e.  _V  A. x  e.  A  ph )
2 ralv 2913 . . 3  |-  ( A. y  e.  _V  ph  <->  A. y ph )
32ralbii 2674 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. x  e.  A  A. y ph )
4 ralv 2913 . 2  |-  ( A. y  e.  _V  A. x  e.  A  ph  <->  A. y A. x  e.  A  ph )
51, 3, 43bitr3i 267 1  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1546   A.wral 2650   _Vcvv 2900
This theorem is referenced by:  uniiunlem  3375  iunss  4074  disjor  4138  trint  4259  reliun  4936  funimass4  5717  ralrnmpt2  6124  findcard3  7287  kmlem12  7975  fimaxre3  9890  vdwmc2  13275  ramtlecl  13296  iunocv  16832  1stccn  17448  itg2leub  19494  nmoubi  22122  nmopub  23260  nmfnleub  23277  disjorf  23866  funcnv5mpt  23926  untuni  24938  mptelee  25549  heibor1lem  26210  ralxpxfr2d  26433  pmapglbx  29884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-v 2902
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