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Theorem eqbrrdv 5013
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 4368 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
3 df-br 4368 . . . 4  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
41, 2, 33bitr3g 287 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
54alrimivv 1728 . 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 5005 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
96, 7, 8syl2anc 659 . 2  |-  ( ph  ->  ( A  =  B  <->  A. x A. y (
<. x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B ) ) )
105, 9mpbird 232 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1397    = wceq 1399    e. wcel 1826   <.cop 3950   class class class wbr 4367   Rel wrel 4918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920
This theorem is referenced by:  eqbrrdva  5085  oppcsect2  15185
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