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Theorem copsex2t 4590
Description: Closed theorem form of copsex2g 4591. (Contributed by NM, 17-Feb-2013.)
Assertion
Ref Expression
copsex2t  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( A  e.  V  /\  B  e.  W
) )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
)
Distinct variable groups:    x, y, ps    x, A, y    x, B, y
Allowed substitution hints:    ph( x, y)    V( x, y)    W( x, y)

Proof of Theorem copsex2t
StepHypRef Expression
1 elisset 2995 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2995 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 566 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1932 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 212 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 nfa1 1831 . . . 4  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps ) )
7 nfe1 1778 . . . . 5  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
8 nfv 1673 . . . . 5  |-  F/ x ps
97, 8nfbi 1867 . . . 4  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
10 nfa2 1879 . . . . 5  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )
11 nfe1 1778 . . . . . . 7  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
1211nfex 1874 . . . . . 6  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
13 nfv 1673 . . . . . 6  |-  F/ y ps
1412, 13nfbi 1867 . . . . 5  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
15 opeq12 4073 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
16 copsexg 4588 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1716eqcoms 2446 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1815, 17syl 16 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1918adantl 466 . . . . . . 7  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( x  =  A  /\  y  =  B
) )  ->  ( ph 
<->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph ) ) )
20 2sp 1801 . . . . . . . 8  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  ( (
x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
) )
2120imp 429 . . . . . . 7  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( x  =  A  /\  y  =  B
) )  ->  ( ph 
<->  ps ) )
2219, 21bitr3d 255 . . . . . 6  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( x  =  A  /\  y  =  B
) )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
)
2322ex 434 . . . . 5  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  ( (
x  =  A  /\  y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) ) )
2410, 14, 23exlimd 1847 . . . 4  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  ( E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
) )
256, 9, 24exlimd 1847 . . 3  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
) )
2625imp 429 . 2  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  E. x E. y ( x  =  A  /\  y  =  B )
)  ->  ( E. x E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
)
275, 26sylan2 474 1  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( A  e.  V  /\  B  e.  W
) )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   <.cop 3895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896
This theorem is referenced by:  opelopabt  4613
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