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Theorem spc2gv 2871
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 285 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( -.  ph  <->  -.  ps )
)
32spc2egv 2870 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  E. x E. y  -. 
ph ) )
4 2nalexn 1560 . . 3  |-  ( -. 
A. x A. y ph 
<->  E. x E. y  -.  ph )
53, 4syl6ibr 218 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  -. 
A. x A. y ph ) )
65con4d 97 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 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684
This theorem is referenced by:  trel  4120  elovmpt2  6064  seqf1olem2  11086  seqf1o  11087  pslem  14315  cnmpt12  17361  cnmpt22  17368  ismrcd2  26774  ismrc  26776  lpolconN  31677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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