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Theorem spc3gv 2886
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 spc3egv.1 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
21notbid 285 . . . 4  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( -.  ph  <->  -.  ps )
)
32spc3egv 2885 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( -.  ps  ->  E. x E. y E. z  -.  ph )
)
4 exnal 1564 . . . . . . 7  |-  ( E. z  -.  ph  <->  -.  A. z ph )
54exbii 1572 . . . . . 6  |-  ( E. y E. z  -. 
ph 
<->  E. y  -.  A. z ph )
6 exnal 1564 . . . . . 6  |-  ( E. y  -.  A. z ph 
<->  -.  A. y A. z ph )
75, 6bitri 240 . . . . 5  |-  ( E. y E. z  -. 
ph 
<->  -.  A. y A. z ph )
87exbii 1572 . . . 4  |-  ( E. x E. y E. z  -.  ph  <->  E. x  -.  A. y A. z ph )
9 exnal 1564 . . . 4  |-  ( E. x  -.  A. y A. z ph  <->  -.  A. x A. y A. z ph )
108, 9bitr2i 241 . . 3  |-  ( -. 
A. x A. y A. z ph  <->  E. x E. y E. z  -. 
ph )
113, 10syl6ibr 218 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( -.  ps  ->  -. 
A. x A. y A. z ph ) )
1211con4d 97 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:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  funopg  5302  pslem  14331  dirtr  14374  fununiq  24197  preotr2  25338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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