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Theorem eqrelriiv 5037
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 5036 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
51, 2, 4mp2an 672 1  |-  A  =  B
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   <.cop 3986   Rel wrel 4948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4454  df-xp 4949  df-rel 4950
This theorem is referenced by:  eqbrriv  5038  inopab  5073  difopab  5074  dfres2  5261  restidsing  5265  cnvopab  5341  cnv0  5343  cnvdif  5346  difxp  5365  cnvcnvsn  5419  dfco2  5440  coiun  5450  co02  5454  coass  5459  ressn  5476  ovoliunlem1  21112  h2hlm  24529  cnvco1  27709  cnvco2  27710
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