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Theorem spc3egv 2698
Description: Existential 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
spc3egv  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ps  ->  E. x E. y E. z ph ) )
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 spc3egv
StepHypRef Expression
1 elisset 2622 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2622 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
3 elisset 2622 . . . 4  |-  ( C  e.  X  ->  E. z 
z  =  C )
41, 2, 33anim123i 1124 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z  z  =  C ) )
5 eeeanv 1851 . . 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 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
87biimprcd 158 . . . 4  |-  ( ps 
->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ph )
)
98eximdv 1803 . . 3  |-  ( ps 
->  ( E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z ph )
)
1092eximdv 1805 . 2  |-  ( ps 
->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. x E. y E. z ph ) )
116, 10syl5com 29 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ps  ->  E. x E. y E. z ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612
This theorem is referenced by: (None)
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