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Theorem predbrg 25453
Description: Closed form of elpredim 25443. (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 25435 . . . . 5  |-  ( x  =  X  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A ,  X ) )
21eleq2d 2502 . . . 4  |-  ( x  =  X  ->  ( Y  e.  Pred ( R ,  A ,  x
)  <->  Y  e.  Pred ( R ,  A ,  X ) ) )
3 breq2 4208 . . . 4  |-  ( x  =  X  ->  ( Y R x  <->  Y R X ) )
42, 3imbi12d 312 . . 3  |-  ( x  =  X  ->  (
( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )  <->  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X ) ) )
5 vex 2951 . . . 4  |-  x  e. 
_V
65elpredim 25443 . . 3  |-  ( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )
74, 6vtoclg 3003 . 2  |-  ( X  e.  V  ->  ( Y  e.  Pred ( R ,  A ,  X
)  ->  Y R X ) )
87imp 419 1  |-  ( ( X  e.  V  /\  Y  e.  Pred ( R ,  A ,  X
) )  ->  Y R X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   Predcpred 25430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-pred 25431
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