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Theorem ralcom4 2703
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 2592 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. y  e.  _V  A. x  e.  A  ph )
2 ralv 2698 . . 3  |-  ( A. y  e.  _V  ph  <->  A. y ph )
32ralbii 2439 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. x  e.  A  A. y ph )
4 ralv 2698 . 2  |-  ( A. y  e.  _V  A. x  e.  A  ph  <->  A. y A. x  e.  A  ph )
51, 3, 43bitr3i 209 1  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329   A.wral 2414   _Vcvv 2681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683
This theorem is referenced by:  uniiunlem  3180  uni0b  3756  iunss  3849  disjnim  3915  trint  4036  reliun  4655  funimass4  5465  ralrnmpo  5878
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