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Theorem spc3gv 3141
Description: Specialization with three 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 296 . . . 4  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( -.  ph  <->  -.  ps )
)
32spc3egv 3140 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( -.  ps  ->  E. x E. y E. z  -.  ph )
)
4 exnal 1701 . . . . . . 7  |-  ( E. z  -.  ph  <->  -.  A. z ph )
54exbii 1720 . . . . . 6  |-  ( E. y E. z  -. 
ph 
<->  E. y  -.  A. z ph )
6 exnal 1701 . . . . . 6  |-  ( E. y  -.  A. z ph 
<->  -.  A. y A. z ph )
75, 6bitri 253 . . . . 5  |-  ( E. y E. z  -. 
ph 
<->  -.  A. y A. z ph )
87exbii 1720 . . . 4  |-  ( E. x E. y E. z  -.  ph  <->  E. x  -.  A. y A. z ph )
9 exnal 1701 . . . 4  |-  ( E. x  -.  A. y A. z ph  <->  -.  A. x A. y A. z ph )
108, 9bitr2i 254 . . 3  |-  ( -. 
A. x A. y A. z ph  <->  E. x E. y E. z  -. 
ph )
113, 10syl6ibr 231 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( -.  ps  ->  -. 
A. x A. y A. z ph ) )
1211con4d 109 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x A. y A. z ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ w3a 986   A.wal 1444    = wceq 1446   E.wex 1665    e. wcel 1889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3049
This theorem is referenced by:  funopg  5617  pslem  16464  dirtr  16494  mclsax  30219  fununiq  30422
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