MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spc2gv Structured version   Unicode version

Theorem spc2gv 3156
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 294 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( -.  ph  <->  -.  ps )
)
32spc2egv 3155 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  E. x E. y  -. 
ph ) )
4 2nalexn 1620 . . 3  |-  ( -. 
A. x A. y ph 
<->  E. x E. y  -.  ph )
53, 4syl6ibr 227 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ps  ->  -. 
A. x A. y ph ) )
65con4d 105 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 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3070
This theorem is referenced by:  trel  4490  elovmpt2  6407  seqf1olem2  11947  seqf1o  11948  brfi1uzind  12321  pslem  15478  cnmpt12  19356  cnmpt22  19363  mbfresfi  28576  ismrcd2  29173  ismrc  29175  frg2woteqm  30790  lpolconN  35438
  Copyright terms: Public domain W3C validator