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Theorem raaanv 3501
Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
Assertion
Ref Expression
raaanv  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) )
Distinct variable groups:    ph, y    ps, x    x, y, A
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem raaanv
StepHypRef Expression
1 nfv 1508 . 2  |-  F/ y
ph
2 nfv 1508 . 2  |-  F/ x ps
31, 2raaan 3500 1  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wral 2435
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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440
This theorem is referenced by:  reusv3i  4420  f1mpt  5722
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