MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqbrriv Unicode version

Theorem eqbrriv 4798
Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Hypotheses
Ref Expression
eqbrriv.1  |-  Rel  A
eqbrriv.2  |-  Rel  B
eqbrriv.3  |-  ( x A y  <->  x B
y )
Assertion
Ref Expression
eqbrriv  |-  A  =  B
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem eqbrriv
StepHypRef Expression
1 eqbrriv.1 . 2  |-  Rel  A
2 eqbrriv.2 . 2  |-  Rel  B
3 eqbrriv.3 . . 3  |-  ( x A y  <->  x B
y )
4 df-br 4040 . . 3  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
5 df-br 4040 . . 3  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
63, 4, 53bitr3i 266 . 2  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
71, 2, 6eqrelriiv 4797 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   Rel wrel 4710
This theorem is referenced by:  resco  5193  tpostpos  6270  sbthcl  6999  dfle2  10497  dflt2  10498  idsset  24501  dfbigcup2  24510  imageval  24540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712
  Copyright terms: Public domain W3C validator