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Theorem brabg2 30181
Description: Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
brabg2.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
brabg2.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
brabg2.3  |-  R  =  { <. x ,  y
>.  |  ph }
brabg2.4  |-  ( ch 
->  A  e.  C
)
Assertion
Ref Expression
brabg2  |-  ( B  e.  D  ->  ( A R B  <->  ch )
)
Distinct variable groups:    x, A, y    x, B, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    C( x, y)    D( x, y)    R( x, y)

Proof of Theorem brabg2
StepHypRef Expression
1 brabg2.3 . . . . 5  |-  R  =  { <. x ,  y
>.  |  ph }
21relopabi 5118 . . . 4  |-  Rel  R
32brrelexi 5030 . . 3  |-  ( A R B  ->  A  e.  _V )
4 brabg2.1 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
5 brabg2.2 . . . . . . 7  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
64, 5, 1brabg 4756 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
76biimpd 207 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( A R B  ->  ch ) )
87ex 434 . . . 4  |-  ( A  e.  _V  ->  ( B  e.  D  ->  ( A R B  ->  ch ) ) )
98com3l 81 . . 3  |-  ( B  e.  D  ->  ( A R B  ->  ( A  e.  _V  ->  ch ) ) )
103, 9mpdi 42 . 2  |-  ( B  e.  D  ->  ( A R B  ->  ch ) )
11 brabg2.4 . . 3  |-  ( ch 
->  A  e.  C
)
124, 5, 1brabg 4756 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
1312exbiri 622 . . . 4  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  A R B ) ) )
1413com3l 81 . . 3  |-  ( B  e.  D  ->  ( ch  ->  ( A  e.  C  ->  A R B ) ) )
1511, 14mpdi 42 . 2  |-  ( B  e.  D  ->  ( ch  ->  A R B ) )
1610, 15impbid 191 1  |-  ( B  e.  D  ->  ( A R B  <->  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   class class class wbr 4437   {copab 4494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996
This theorem is referenced by: (None)
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