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

Theorem eqbrrdiv 4718
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 3999 . . 3  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
5 df-br 3999 . . 3  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
63, 4, 53bitr3g 222 . 2  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
71, 2, 6eqrelrdv 4716 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2146   <.cop 3592   class class class wbr 3998   Rel wrel 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator