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Theorem ralcom4 2806
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 2700 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. y  e.  _V  A. x  e.  A  ph )
2 ralv 2801 . . 3  |-  ( A. y  e.  _V  ph  <->  A. y ph )
32ralbii 2567 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. x  e.  A  A. y ph )
4 ralv 2801 . 2  |-  ( A. y  e.  _V  A. x  e.  A  ph  <->  A. y A. x  e.  A  ph )
51, 3, 43bitr3i 266 1  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   A.wral 2543   _Vcvv 2788
This theorem is referenced by:  uniiunlem  3260  iunss  3943  disjor  4007  trint  4128  reliun  4806  funimass4  5573  ralrnmpt2  5958  findcard3  7100  kmlem12  7787  fimaxre3  9703  vdwmc2  13026  ramtlecl  13047  iunocv  16581  1stccn  17189  itg2leub  19089  nmoubi  21350  nmopub  22488  nmfnleub  22505  funcnv5mpt  23236  untuni  24055  mptelee  24523  heibor1lem  26533  ralxpxfr2d  26760  pmapglbx  29958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790
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