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Theorem predbrg 24186
Description: Closed form of elpredim 24176. (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 24171 . . . . 5  |-  ( x  =  X  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A ,  X ) )
21eleq2d 2350 . . . 4  |-  ( x  =  X  ->  ( Y  e.  Pred ( R ,  A ,  x
)  <->  Y  e.  Pred ( R ,  A ,  X ) ) )
3 breq2 4027 . . . 4  |-  ( x  =  X  ->  ( Y R x  <->  Y R X ) )
42, 3imbi12d 311 . . 3  |-  ( x  =  X  ->  (
( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )  <->  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X ) ) )
5 vex 2791 . . . 4  |-  x  e. 
_V
65elpredim 24176 . . 3  |-  ( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )
74, 6vtoclg 2843 . 2  |-  ( X  e.  V  ->  ( Y  e.  Pred ( R ,  A ,  X
)  ->  Y R X ) )
87imp 418 1  |-  ( ( X  e.  V  /\  Y  e.  Pred ( R ,  A ,  X
) )  ->  Y R X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   Predcpred 24167
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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