ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opelopabt Unicode version

Theorem opelopabt 4025
Description: Closed theorem form of opelopab 4034. (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 4021 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) )
2 19.26-2 1412 . . . . 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 1386 . . . . 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 4008 . . . 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 1134 . 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 920   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   <.cop 3409   {copab 3846
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator