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Theorem bnj1148 29083
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 2934 . . . . 5  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 453 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
3 bnj93 28952 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
4 eleq1 2472 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54anbi2d 685 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
6 bnj602 29004 . . . . . . 7  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
76eleq1d 2478 . . . . . 6  |-  ( x  =  X  ->  (  pred ( x ,  A ,  R )  e.  _V  <->  pred ( X ,  A ,  R )  e.  _V ) )
85, 7imbi12d 312 . . . . 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 203 . . . 4  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
102, 9bnj593 28831 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V ) )
1110bnj937 28860 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( ( R  FrSe  A  /\  X  e.  A
)  ->  pred ( X ,  A ,  R
)  e.  _V )
)
1211pm2.43i 45 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2924    predc-bnj14 28770    FrSe w-bnj15 28774
This theorem is referenced by:  bnj1136  29084  bnj1413  29122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-bnj14 28771  df-bnj13 28773  df-bnj15 28775
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