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Theorem rspc3v 2893
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 2563 . . . 4  |-  ( x  =  A  ->  ( A. z  e.  T  ph  <->  A. z  e.  T  ch ) )
3 rspc3v.2 . . . . 5  |-  ( y  =  B  ->  ( ch 
<->  th ) )
43ralbidv 2563 . . . 4  |-  ( y  =  B  ->  ( A. z  e.  T  ch 
<-> 
A. z  e.  T  th ) )
52, 4rspc2v 2890 . . 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 2880 . . 3  |-  ( C  e.  T  ->  ( A. z  e.  T  th  ->  ps ) )
85, 7sylan9 638 . 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 1146 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543
This theorem is referenced by:  swopolem  4323  isopolem  5842  caovassg  6018  caovcang  6021  caovordig  6025  caovordg  6027  caovdig  6034  caovdirg  6037  caofass  6111  caoftrn  6112  prslem  14065  posi  14084  latdisdlem  14292  dlatmjdi  14297  mndlem1  14371  gaass  14751  islmodd  15633  lsscl  15700  assalem  16057  xmettri2  17905  grpoass  20870  isgrp2d  20902  rngodi  21052  rngodir  21053  rngoass  21054  vcdi  21108  vcdir  21109  vcass  21110  lnolin  21332  lnopl  22494  lnfnl  22511  smgrpass2  25341  mndoass2  25360  vecax5b  25459  glmrngo  25482  vecax5c  25483  cmpasso  25773  lfli  29251  cvlexch1  29518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790
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