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Theorem spc3egv 2872
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 2798 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2798 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
3 elisset 2798 . . . 4  |-  ( C  e.  X  ->  E. z 
z  =  C )
41, 2, 33anim123i 1137 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z  z  =  C ) )
5 eeeanv 1855 . . 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 203 . 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 216 . . . 4  |-  ( ps 
->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ph )
)
98eximdv 1608 . . 3  |-  ( ps 
->  ( E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z ph )
)
1092eximdv 1610 . 2  |-  ( ps 
->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. x E. y E. z ph ) )
116, 10syl5com 26 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 176    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684
This theorem is referenced by:  spc3gv  2873  dihjatcclem4  31611
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-ext 2264
This theorem depends on definitions:  df-bi 177  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-v 2790
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