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Theorem predbrg 28843
Description: Closed form of elpredim 28833. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)
Assertion
Ref Expression
predbrg  |-  ( ( X  e.  V  /\  Y  e.  Pred ( R ,  A ,  X
) )  ->  Y R X )

Proof of Theorem predbrg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 predeq3 28825 . . . . 5  |-  ( x  =  X  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A ,  X ) )
21eleq2d 2537 . . . 4  |-  ( x  =  X  ->  ( Y  e.  Pred ( R ,  A ,  x
)  <->  Y  e.  Pred ( R ,  A ,  X ) ) )
3 breq2 4451 . . . 4  |-  ( x  =  X  ->  ( Y R x  <->  Y R X ) )
42, 3imbi12d 320 . . 3  |-  ( x  =  X  ->  (
( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )  <->  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X ) ) )
5 vex 3116 . . . 4  |-  x  e. 
_V
65elpredim 28833 . . 3  |-  ( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )
74, 6vtoclg 3171 . 2  |-  ( X  e.  V  ->  ( Y  e.  Pred ( R ,  A ,  X
)  ->  Y R X ) )
87imp 429 1  |-  ( ( X  e.  V  /\  Y  e.  Pred ( R ,  A ,  X
) )  ->  Y R X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   Predcpred 28820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-pred 28821
This theorem is referenced by: (None)
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