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Theorem cbvral2v 2900
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 2686 . . 3  |-  ( x  =  z  ->  ( A. y  e.  B  ph  <->  A. y  e.  B  ch ) )
32cbvralv 2892 . 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 2892 . . 3  |-  ( A. y  e.  B  ch  <->  A. w  e.  B  ps )
65ralbii 2690 . 2  |-  ( A. z  e.  A  A. y  e.  B  ch  <->  A. z  e.  A  A. w  e.  B  ps )
73, 6bitri 241 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 177   A.wral 2666
This theorem is referenced by:  cbvral3v  2902  fununi  5476  fiint  7342  nqereu  8762  mhmpropd  14699  efgred  15335  fbun  17825  fbunfip  17854  caucfil  19189  pmltpc  19300  ghgrplem2  21908  htth  22374  cdj3lem3b  23896  cdj3i  23897  nofulllem5  25574  axcontlem10  25816  frgrawopreglem5  28151
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671
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