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Theorem spc2gv 3166
Description: Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
Hypothesis
Ref Expression
spc2egv.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
spc2gv  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x A. y ph  ->  ps )
)
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)    V( x, y)    W( x, y)

Proof of Theorem spc2gv
StepHypRef Expression
1 spc2egv.1 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
21notbid 295 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( -.  ph  <->  -.  ps )
)
32spc2egv 3165 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  E. x E. y  -. 
ph ) )
4 2nalexn 1696 . . 3  |-  ( -. 
A. x A. y ph 
<->  E. x E. y  -.  ph )
53, 4syl6ibr 230 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  -. 
A. x A. y ph ) )
65con4d 108 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x A. y ph  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-v 3080
This theorem is referenced by:  trel  4519  elovmpt2  6520  seqf1olem2  12246  seqf1o  12247  brfi1uzind  12637  pslem  16430  cnmpt12  20659  cnmpt22  20666  frg2woteqm  25763  mclsppslem  30210  mbfresfi  31895  lpolconN  34968  ismrcd2  35454  ismrc  35456
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