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Theorem copsex2t 4385
Description: Closed theorem form of copsex2g 4386. (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 2910 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2910 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 550 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1926 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 204 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 nfa1 1796 . . . 4  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps ) )
7 nfe1 1739 . . . . 5  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
8 nfv 1626 . . . . 5  |-  F/ x ps
97, 8nfbi 1846 . . . 4  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
10 nfa2 1867 . . . . 5  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )
11 nfe1 1739 . . . . . . 7  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
1211nfex 1855 . . . . . 6  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
13 nfv 1626 . . . . . 6  |-  F/ y ps
1412, 13nfbi 1846 . . . . 5  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
15 opeq12 3929 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
16 copsexg 4384 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1716eqcoms 2391 . . . . . . . . 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 453 . . . . . . 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 sp 1755 . . . . . . . . 9  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) ) )
212019.21bi 1766 . . . . . . . 8  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  ( (
x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
) )
2221imp 419 . . . . . . 7  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( x  =  A  /\  y  =  B
) )  ->  ( ph 
<->  ps ) )
2319, 22bitr3d 247 . . . . . 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 )
)
2423ex 424 . . . . 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 ) ) )
2510, 14, 24exlimd 1814 . . . 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 )
) )
266, 9, 25exlimd 1814 . . 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 )
) )
2726imp 419 . 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 )
)
285, 27sylan2 461 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   <.cop 3761
This theorem is referenced by:  opelopabt  4409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767
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