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Theorem spc2gv 3031
Description: Specialization with 2 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 286 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( -.  ph  <->  -.  ps )
)
32spc2egv 3030 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  E. x E. y  -. 
ph ) )
4 2nalexn 1582 . . 3  |-  ( -. 
A. x A. y ph 
<->  E. x E. y  -.  ph )
53, 4syl6ibr 219 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  -. 
A. x A. y ph ) )
65con4d 99 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x A. y ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725
This theorem is referenced by:  trel  4301  elovmpt2  6283  seqf1olem2  11353  seqf1o  11354  brfi1uzind  11705  pslem  14628  cnmpt12  17689  cnmpt22  17696  mbfresfi  26216  ismrcd2  26707  ismrc  26709  frg2woteqm  28349  lpolconN  32186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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