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Theorem copsex2t 4707
Description: Closed theorem form of copsex2g 4708. (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 3091 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 3091 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 568 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 2047 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 215 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 nfa1 1956 . . . 4  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps ) )
7 nfe1 1894 . . . . 5  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
8 nfv 1755 . . . . 5  |-  F/ x ps
97, 8nfbi 1994 . . . 4  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
10 nfa2 2013 . . . . 5  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )
11 nfe1 1894 . . . . . . 7  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
1211nfex 2008 . . . . . 6  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
13 nfv 1755 . . . . . 6  |-  F/ y ps
1412, 13nfbi 1994 . . . . 5  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
15 opeq12 4189 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
16 copsexg 4706 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1716eqcoms 2434 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1815, 17syl 17 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1918adantl 467 . . . . . . 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 1921 . . . . . . . 8  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  ( (
x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
) )
2120imp 430 . . . . . . 7  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( x  =  A  /\  y  =  B
) )  ->  ( ph 
<->  ps ) )
2219, 21bitr3d 258 . . . . . 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 435 . . . . 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 1974 . . . 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 1974 . . 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 430 . 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 476 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 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   <.cop 4004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005
This theorem is referenced by:  opelopabt  4732
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