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Theorem spc3gv 2711
Description: Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
Hypothesis
Ref Expression
spc3egv.1  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
spc3gv  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x A. y A. z ph  ->  ps ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    ps, x, y, z
Allowed substitution hints:    ph( x, y, z)    V( x, y, z)    W( x, y, z)    X( x, y, z)

Proof of Theorem spc3gv
StepHypRef Expression
1 elisset 2633 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2633 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
3 elisset 2633 . . . 4  |-  ( C  e.  X  ->  E. z 
z  =  C )
41, 2, 33anim123i 1128 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z  z  =  C ) )
5 eeeanv 1856 . . 3  |-  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  <->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z 
z  =  C ) )
64, 5sylibr 132 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C ) )
7 spc3egv.1 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
87biimpcd 157 . . . . . . 7  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ps ) )
982alimi 1390 . . . . . 6  |-  ( A. y A. z ph  ->  A. y A. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ps )
)
109alimi 1389 . . . . 5  |-  ( A. x A. y A. z ph  ->  A. x A. y A. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ps ) )
11 exim 1535 . . . . . 6  |-  ( A. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ps )  ->  ( E. z
( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z ps ) )
12112alimi 1390 . . . . 5  |-  ( A. x A. y A. z
( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ps )  ->  A. x A. y
( E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z ps )
)
1310, 12syl 14 . . . 4  |-  ( A. x A. y A. z ph  ->  A. x A. y
( E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z ps )
)
14 exim 1535 . . . . 5  |-  ( A. y ( E. z
( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z ps )  ->  ( E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. y E. z ps ) )
1514alimi 1389 . . . 4  |-  ( A. x A. y ( E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z ps )  ->  A. x
( E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. y E. z ps ) )
16 exim 1535 . . . 4  |-  ( A. x ( E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. y E. z ps )  -> 
( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. x E. y E. z ps ) )
1713, 15, 163syl 17 . . 3  |-  ( A. x A. y A. z ph  ->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. x E. y E. z ps ) )
18 19.9v 1799 . . . 4  |-  ( E. x E. y E. z ps  <->  E. y E. z ps )
19 19.9v 1799 . . . 4  |-  ( E. y E. z ps  <->  E. z ps )
20 19.9v 1799 . . . 4  |-  ( E. z ps  <->  ps )
2118, 19, 203bitri 204 . . 3  |-  ( E. x E. y E. z ps  <->  ps )
2217, 21syl6ib 159 . 2  |-  ( A. x A. y A. z ph  ->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ps ) )
236, 22syl5com 29 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x A. y A. z ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 924   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  funopg  5034
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