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

Theorem eqbrrdiv 4484
Description: Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Hypotheses
Ref Expression
eqbrrdiv.1  |-  Rel  A
eqbrrdiv.2  |-  Rel  B
eqbrrdiv.3  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
Assertion
Ref Expression
eqbrrdiv  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, y, A   
x, B, y    ph, x, y

Proof of Theorem eqbrrdiv
StepHypRef Expression
1 eqbrrdiv.1 . 2  |-  Rel  A
2 eqbrrdiv.2 . 2  |-  Rel  B
3 eqbrrdiv.3 . . 3  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
4 df-br 3806 . . 3  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
5 df-br 3806 . . 3  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
63, 4, 53bitr3g 220 . 2  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
71, 2, 6eqrelrdv 4482 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   <.cop 3419   class class class wbr 3805   Rel wrel 4396
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 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator