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Theorem spc2gv 2709
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 elisset 2633 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2633 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 331 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1855 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 132 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 spc2egv.1 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
76biimpcd 157 . . . . 5  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ps ) )
872alimi 1390 . . . 4  |-  ( A. x A. y ph  ->  A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ps )
)
9 exim 1535 . . . . 5  |-  ( A. y ( ( x  =  A  /\  y  =  B )  ->  ps )  ->  ( E. y
( x  =  A  /\  y  =  B )  ->  E. y ps ) )
109alimi 1389 . . . 4  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ps )  ->  A. x
( E. y ( x  =  A  /\  y  =  B )  ->  E. y ps )
)
11 exim 1535 . . . 4  |-  ( A. x ( E. y
( x  =  A  /\  y  =  B )  ->  E. y ps )  ->  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y ps ) )
128, 10, 113syl 17 . . 3  |-  ( A. x A. y ph  ->  ( E. x E. y
( x  =  A  /\  y  =  B )  ->  E. x E. y ps ) )
13 19.9v 1799 . . . 4  |-  ( E. x E. y ps  <->  E. y ps )
14 19.9v 1799 . . . 4  |-  ( E. y ps  <->  ps )
1513, 14bitri 182 . . 3  |-  ( E. x E. y ps  <->  ps )
1612, 15syl6ib 159 . 2  |-  ( A. x A. y ph  ->  ( E. x E. y
( x  =  A  /\  y  =  B )  ->  ps )
)
175, 16syl5com 29 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x A. y ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  rspc2gv  2732  trel  3935  exmidundif  4026  elovmpt2  5827  exmidsbthrlem  11569
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