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Theorem brabg2 25515
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 4848 . . . 4  |-  Rel  R
32brrelexi 4766 . . 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 4321 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
76biimpd 198 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( A R B  ->  ch ) )
87ex 423 . . . 4  |-  ( A  e.  _V  ->  ( B  e.  D  ->  ( A R B  ->  ch ) ) )
98com3l 75 . . 3  |-  ( B  e.  D  ->  ( A R B  ->  ( A  e.  _V  ->  ch ) ) )
103, 9mpdi 38 . 2  |-  ( B  e.  D  ->  ( A R B  ->  ch ) )
11 brabg2.4 . . 3  |-  ( ch 
->  A  e.  C
)
124, 5, 1brabg 4321 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
1312exbiri 605 . . . 4  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  A R B ) ) )
1413com3l 75 . . 3  |-  ( B  e.  D  ->  ( ch  ->  ( A  e.  C  ->  A R B ) ) )
1511, 14mpdi 38 . 2  |-  ( B  e.  D  ->  ( ch  ->  A R B ) )
1610, 15impbid 183 1  |-  ( B  e.  D  ->  ( A R B  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822   class class class wbr 4060   {copab 4113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733
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