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Theorem bnj1148 29590
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 3089 . . . . 5  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 467 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
3 bnj93 29459 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
4 eleq1 2492 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54anbi2d 708 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
6 bnj602 29511 . . . . . . 7  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
76eleq1d 2489 . . . . . 6  |-  ( x  =  X  ->  (  pred ( x ,  A ,  R )  e.  _V  <->  pred ( X ,  A ,  R )  e.  _V ) )
85, 7imbi12d 321 . . . . 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 214 . . . 4  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
102, 9bnj593 29340 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V ) )
1110bnj937 29368 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( ( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
1211pm2.43i 49 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   _Vcvv 3078    predc-bnj14 29278    FrSe w-bnj15 29282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-bnj14 29279  df-bnj13 29281  df-bnj15 29283
This theorem is referenced by:  bnj1136  29591  bnj1413  29629
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