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Theorem copsex2t 4743
Description: Closed theorem form of copsex2g 4744. (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 3120 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 3120 . . . 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 1989 . . 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 1898 . . . 4  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps ) )
7 nfe1 1841 . . . . 5  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
8 nfv 1708 . . . . 5  |-  F/ x ps
97, 8nfbi 1935 . . . 4  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
10 nfa2 1954 . . . . 5  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )
11 nfe1 1841 . . . . . . 7  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
1211nfex 1949 . . . . . 6  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
13 nfv 1708 . . . . . 6  |-  F/ y ps
1412, 13nfbi 1935 . . . . 5  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
15 opeq12 4221 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
16 copsexg 4741 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1716eqcoms 2469 . . . . . . . . 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 1867 . . . . . . . 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 1915 . . . 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 1915 . . 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 1393    = wceq 1395   E.wex 1613    e. wcel 1819   <.cop 4038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039
This theorem is referenced by:  opelopabt  4768
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