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Theorem opelopabt 4080
Description: Closed theorem form of opelopab 4089. (Contributed by NM, 19-Feb-2013.)
Assertion
Ref Expression
opelopabt  |-  ( ( A. x A. y
( x  =  A  ->  ( ph  <->  ps )
)  /\  A. x A. y ( y  =  B  ->  ( ps  <->  ch ) )  /\  ( A  e.  V  /\  B  e.  W )
)  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
)
Distinct variable groups:    x, y, A   
x, B, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    V( x, y)    W( x, y)

Proof of Theorem opelopabt
StepHypRef Expression
1 elopab 4076 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) )
2 19.26-2 1416 . . . . 5  |-  ( A. x A. y ( ( x  =  A  -> 
( ph  <->  ps ) )  /\  ( y  =  B  ->  ( ps  <->  ch )
) )  <->  ( A. x A. y ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A. x A. y ( y  =  B  -> 
( ps  <->  ch )
) ) )
3 prth 336 . . . . . . 7  |-  ( ( ( x  =  A  ->  ( ph  <->  ps )
)  /\  ( y  =  B  ->  ( ps  <->  ch ) ) )  -> 
( ( x  =  A  /\  y  =  B )  ->  (
( ph  <->  ps )  /\  ( ps 
<->  ch ) ) ) )
4 bitr 456 . . . . . . 7  |-  ( ( ( ph  <->  ps )  /\  ( ps  <->  ch )
)  ->  ( ph  <->  ch ) )
53, 4syl6 33 . . . . . 6  |-  ( ( ( x  =  A  ->  ( ph  <->  ps )
)  /\  ( y  =  B  ->  ( ps  <->  ch ) ) )  -> 
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ch ) ) )
652alimi 1390 . . . . 5  |-  ( A. x A. y ( ( x  =  A  -> 
( ph  <->  ps ) )  /\  ( y  =  B  ->  ( ps  <->  ch )
) )  ->  A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ch ) ) )
72, 6sylbir 133 . . . 4  |-  ( ( A. x A. y
( x  =  A  ->  ( ph  <->  ps )
)  /\  A. x A. y ( y  =  B  ->  ( ps  <->  ch ) ) )  ->  A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch ) ) )
8 copsex2t 4063 . . . 4  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  ( ph 
<->  ch ) )  /\  ( A  e.  V  /\  B  e.  W
) )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ch )
)
97, 8sylan 277 . . 3  |-  ( ( ( A. x A. y ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x A. y ( y  =  B  ->  ( ps 
<->  ch ) ) )  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ch ) )
1093impa 1138 . 2  |-  ( ( A. x A. y
( x  =  A  ->  ( ph  <->  ps )
)  /\  A. x A. y ( y  =  B  ->  ( ps  <->  ch ) )  /\  ( A  e.  V  /\  B  e.  W )
)  ->  ( E. x E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ch )
)
111, 10syl5bb 190 1  |-  ( ( A. x A. y
( x  =  A  ->  ( ph  <->  ps )
)  /\  A. x A. y ( y  =  B  ->  ( ps  <->  ch ) )  /\  ( A  e.  V  /\  B  e.  W )
)  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3444   {copab 3890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892
This theorem is referenced by: (None)
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