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Theorem opelopabt 4154
Description: Closed theorem form of opelopab 4163. (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 4150 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) )
2 19.26-2 1443 . . . . 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 anim12 341 . . . . . . 7  |-  ( ( ( x  =  A  ->  ( ph  <->  ps )
)  /\  ( y  =  B  ->  ( ps  <->  ch ) ) )  -> 
( ( x  =  A  /\  y  =  B )  ->  (
( ph  <->  ps )  /\  ( ps 
<->  ch ) ) ) )
4 bitr 463 . . . . . . 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 1417 . . . . 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 134 . . . 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 4137 . . . 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 281 . . 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 1161 . 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 191 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 103    <-> wb 104    /\ w3a 947   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465   <.cop 3500   {copab 3958
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960
This theorem is referenced by: (None)
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