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Theorem bnj1148 29463
Description: Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1148  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )

Proof of Theorem bnj1148
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elisset 2972 . . . . 5  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 454 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
3 bnj93 29332 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
4 eleq1 2502 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54anbi2d 686 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
6 bnj602 29384 . . . . . . 7  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
76eleq1d 2508 . . . . . 6  |-  ( x  =  X  ->  (  pred ( x ,  A ,  R )  e.  _V  <->  pred ( X ,  A ,  R )  e.  _V ) )
85, 7imbi12d 313 . . . . 5  |-  ( x  =  X  ->  (
( ( R  FrSe  A  /\  x  e.  A
)  ->  pred ( x ,  A ,  R
)  e.  _V )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
) )
93, 8mpbii 204 . . . 4  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
102, 9bnj593 29211 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V ) )
1110bnj937 29240 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( ( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
1211pm2.43i 46 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1727   _Vcvv 2962    predc-bnj14 29150    FrSe w-bnj15 29154
This theorem is referenced by:  bnj1136  29464  bnj1413  29502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-bnj14 29151  df-bnj13 29153  df-bnj15 29155
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