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Theorem eqbrrdv 5098
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 4448 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
3 df-br 4448 . . . 4  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
41, 2, 33bitr3g 287 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
54alrimivv 1696 . 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 5090 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
96, 7, 8syl2anc 661 . 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 1377    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006
This theorem is referenced by:  eqbrrdva  5170  oppcsect2  15026
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