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Theorem copsex2t 4174
Description: Closed theorem form of copsex2g 4175. (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 2703 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2703 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 336 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1905 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 133 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 nfa1 1522 . . . 4  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps ) )
7 nfe1 1473 . . . . 5  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
8 nfv 1509 . . . . 5  |-  F/ x ps
97, 8nfbi 1569 . . . 4  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
10 nfa2 1559 . . . . 5  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )
11 nfe1 1473 . . . . . . 7  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
1211nfex 1617 . . . . . 6  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
13 nfv 1509 . . . . . 6  |-  F/ y ps
1412, 13nfbi 1569 . . . . 5  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
15 opeq12 3714 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
16 copsexg 4173 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1716eqcoms 2143 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1815, 17syl 14 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1918adantl 275 . . . . . . 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 1489 . . . . . . . . 9  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) ) )
212019.21bi 1538 . . . . . . . 8  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)  ->  ( (
x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
) )
2221imp 123 . . . . . . 7  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ps ) )  /\  ( x  =  A  /\  y  =  B
) )  ->  ( ph 
<->  ps ) )
2319, 22bitr3d 189 . . . . . 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 114 . . . . 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 1577 . . . 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 1577 . . 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 123 . 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 284 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    /\ wa 103    <-> wb 104   A.wal 1330    = wceq 1332   E.wex 1469    e. wcel 1481   <.cop 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540
This theorem is referenced by:  opelopabt  4191
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