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Theorem eqbrrdv 4931
Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
eqbrrdv.1  |-  ( ph  ->  Rel  A )
eqbrrdv.2  |-  ( ph  ->  Rel  B )
eqbrrdv.3  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
Assertion
Ref Expression
eqbrrdv  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqbrrdv
StepHypRef Expression
1 eqbrrdv.3 . . . 4  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
2 df-br 4402 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
3 df-br 4402 . . . 4  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
41, 2, 33bitr3g 291 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
54alrimivv 1773 . 2  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
6 eqbrrdv.1 . . 3  |-  ( ph  ->  Rel  A )
7 eqbrrdv.2 . . 3  |-  ( ph  ->  Rel  B )
8 eqrel 4923 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
96, 7, 8syl2anc 666 . 2  |-  ( ph  ->  ( A  =  B  <->  A. x A. y (
<. x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B ) ) )
105, 9mpbird 236 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1441    = wceq 1443    e. wcel 1886   <.cop 3973   class class class wbr 4401   Rel wrel 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-xp 4839  df-rel 4840
This theorem is referenced by:  eqbrrdva  5003  oppcsect2  15677
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