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Theorem eqrelriiv 4603
Description: Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
Hypotheses
Ref Expression
eqreliiv.1  |-  Rel  A
eqreliiv.2  |-  Rel  B
eqreliiv.3  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
Assertion
Ref Expression
eqrelriiv  |-  A  =  B
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem eqrelriiv
StepHypRef Expression
1 eqreliiv.1 . 2  |-  Rel  A
2 eqreliiv.2 . 2  |-  Rel  B
3 eqreliiv.3 . . 3  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
43eqrelriv 4602 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
51, 2, 4mp2an 422 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316    e. wcel 1465   <.cop 3500   Rel wrel 4514
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-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  df-xp 4515  df-rel 4516
This theorem is referenced by:  eqbrriv  4604  inopab  4641  difopab  4642  dfres2  4841  cnvopab  4910  cnv0  4912  cnvdif  4915  cnvcnvsn  4985  dfco2  5008  coiun  5018  co02  5022  coass  5027  ressn  5049
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