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Theorem predbrg 27647
Description: Closed form of elpredim 27637. (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 27629 . . . . 5  |-  ( x  =  X  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A ,  X ) )
21eleq2d 2510 . . . 4  |-  ( x  =  X  ->  ( Y  e.  Pred ( R ,  A ,  x
)  <->  Y  e.  Pred ( R ,  A ,  X ) ) )
3 breq2 4296 . . . 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 2975 . . . 4  |-  x  e. 
_V
65elpredim 27637 . . 3  |-  ( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )
74, 6vtoclg 3030 . 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 1369    e. wcel 1756   class class class wbr 4292   Predcpred 27624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-cnv 4848  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-pred 27625
This theorem is referenced by: (None)
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