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Theorem bnj1148 34153
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 3120 . . . . 5  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 466 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
3 bnj93 34022 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
4 eleq1 2529 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54anbi2d 703 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
6 bnj602 34074 . . . . . . 7  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
76eleq1d 2526 . . . . . 6  |-  ( x  =  X  ->  (  pred ( x ,  A ,  R )  e.  _V  <->  pred ( X ,  A ,  R )  e.  _V ) )
85, 7imbi12d 320 . . . . 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 211 . . . 4  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
102, 9bnj593 33903 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V ) )
1110bnj937 33931 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( ( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
1211pm2.43i 47 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   _Vcvv 3109    predc-bnj14 33841    FrSe w-bnj15 33845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-bnj14 33842  df-bnj13 33844  df-bnj15 33846
This theorem is referenced by:  bnj1136  34154  bnj1413  34192
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