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Theorem cbvral3v 2639
Description: Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
Hypotheses
Ref Expression
cbvral3v.1  |-  ( x  =  w  ->  ( ph 
<->  ch ) )
cbvral3v.2  |-  ( y  =  v  ->  ( ch 
<->  th ) )
cbvral3v.3  |-  ( z  =  u  ->  ( th 
<->  ps ) )
Assertion
Ref Expression
cbvral3v  |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  <->  A. w  e.  A  A. v  e.  B  A. u  e.  C  ps )
Distinct variable groups:    ph, w    ps, z    ch, x    ch, v    y, u, th    x, A    w, A    x, y, B   
y, w, B    v, B    x, z, C, y   
z, w, C    z,
v, C    u, C
Allowed substitution hints:    ph( x, y, z, v, u)    ps( x, y, w, v, u)    ch( y, z, w, u)    th( x, z, w, v)    A( y, z, v, u)    B( z, u)

Proof of Theorem cbvral3v
StepHypRef Expression
1 cbvral3v.1 . . . 4  |-  ( x  =  w  ->  ( ph 
<->  ch ) )
212ralbidv 2434 . . 3  |-  ( x  =  w  ->  ( A. y  e.  B  A. z  e.  C  ph  <->  A. y  e.  B  A. z  e.  C  ch ) )
32cbvralv 2629 . 2  |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  <->  A. w  e.  A  A. y  e.  B  A. z  e.  C  ch )
4 cbvral3v.2 . . . 4  |-  ( y  =  v  ->  ( ch 
<->  th ) )
5 cbvral3v.3 . . . 4  |-  ( z  =  u  ->  ( th 
<->  ps ) )
64, 5cbvral2v 2637 . . 3  |-  ( A. y  e.  B  A. z  e.  C  ch  <->  A. v  e.  B  A. u  e.  C  ps )
76ralbii 2416 . 2  |-  ( A. w  e.  A  A. y  e.  B  A. z  e.  C  ch  <->  A. w  e.  A  A. v  e.  B  A. u  e.  C  ps )
83, 7bitri 183 1  |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  <->  A. w  e.  A  A. v  e.  B  A. u  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wral 2391
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396
This theorem is referenced by: (None)
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