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Theorem cbvral2v 2598
Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
Hypotheses
Ref Expression
cbvral2v.1  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
cbvral2v.2  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
cbvral2v  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
Distinct variable groups:    x, A    z, A    x, y, B    y,
z, B    w, B    ph, z    ps, y    ch, x    ch, w
Allowed substitution hints:    ph( x, y, w)    ps( x, z, w)    ch( y, z)    A( y, w)

Proof of Theorem cbvral2v
StepHypRef Expression
1 cbvral2v.1 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
21ralbidv 2380 . . 3  |-  ( x  =  z  ->  ( A. y  e.  B  ph  <->  A. y  e.  B  ch ) )
32cbvralv 2590 . 2  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. y  e.  B  ch )
4 cbvral2v.2 . . . 4  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
54cbvralv 2590 . . 3  |-  ( A. y  e.  B  ch  <->  A. w  e.  B  ps )
65ralbii 2384 . 2  |-  ( A. z  e.  A  A. y  e.  B  ch  <->  A. z  e.  A  A. w  e.  B  ps )
73, 6bitri 182 1  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wral 2359
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-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364
This theorem is referenced by:  cbvral3v  2600  fununi  5076  fiintim  6629  isoti  6692  cauappcvgprlemlim  7210  caucvgprlemnkj  7215  caucvgprlemcl  7225  caucvgprprlemcbv  7236  axcaucvglemcau  7423  iseqdistr  9933  seq3distr  9934
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