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Theorem rspc3v 2724
Description: 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
Hypotheses
Ref Expression
rspc3v.1  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
rspc3v.2  |-  ( y  =  B  ->  ( ch 
<->  th ) )
rspc3v.3  |-  ( z  =  C  ->  ( th 
<->  ps ) )
Assertion
Ref Expression
rspc3v  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  ( A. x  e.  R  A. y  e.  S  A. z  e.  T  ph  ->  ps ) )
Distinct variable groups:    ps, z    ch, x    th, y    x, y, z, A    y, B, z    z, C    x, R    x, S, y    x, T, y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y)    ch( y,
z)    th( x, z)    B( x)    C( x, y)    R( y, z)    S( z)

Proof of Theorem rspc3v
StepHypRef Expression
1 rspc3v.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
21ralbidv 2373 . . . 4  |-  ( x  =  A  ->  ( A. z  e.  T  ph  <->  A. z  e.  T  ch ) )
3 rspc3v.2 . . . . 5  |-  ( y  =  B  ->  ( ch 
<->  th ) )
43ralbidv 2373 . . . 4  |-  ( y  =  B  ->  ( A. z  e.  T  ch 
<-> 
A. z  e.  T  th ) )
52, 4rspc2v 2721 . . 3  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A. x  e.  R  A. y  e.  S  A. z  e.  T  ph  ->  A. z  e.  T  th )
)
6 rspc3v.3 . . . 4  |-  ( z  =  C  ->  ( th 
<->  ps ) )
76rspcv 2706 . . 3  |-  ( C  e.  T  ->  ( A. z  e.  T  th  ->  ps ) )
85, 7sylan9 401 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  C  e.  T )  ->  ( A. x  e.  R  A. y  e.  S  A. z  e.  T  ph 
->  ps ) )
983impa 1134 1  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  ( A. x  e.  R  A. y  e.  S  A. z  e.  T  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612
This theorem is referenced by:  swopolem  4088  isopolem  5512  isosolem  5514  caovassg  5710  caovcang  5713  caovordig  5717  caovordg  5719  caovdig  5726  caovdirg  5729  caoftrn  5787
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