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Theorem bnj1148 29026
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 2798 . . . . 5  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 452 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
3 bnj93 28895 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
4 eleq1 2343 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54anbi2d 684 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
6 bnj602 28947 . . . . . . 7  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
76eleq1d 2349 . . . . . 6  |-  ( x  =  X  ->  (  pred ( x ,  A ,  R )  e.  _V  <->  pred ( X ,  A ,  R )  e.  _V ) )
85, 7imbi12d 311 . . . . 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 202 . . . 4  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
102, 9bnj593 28774 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V ) )
1110bnj937 28803 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( ( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
1211pm2.43i 43 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj1136  29027  bnj1413  29065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718
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