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Theorem predbrg 5389
Description: Closed form of elpredim 5381. (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 5373 . . . . 5  |-  ( x  =  X  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A ,  X ) )
21eleq2d 2474 . . . 4  |-  ( x  =  X  ->  ( Y  e.  Pred ( R ,  A ,  x
)  <->  Y  e.  Pred ( R ,  A ,  X ) ) )
3 breq2 4401 . . . 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 3064 . . . 4  |-  x  e. 
_V
65elpredim 5381 . . 3  |-  ( Y  e.  Pred ( R ,  A ,  x )  ->  Y R x )
74, 6vtoclg 3119 . 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 1407    e. wcel 1844   class class class wbr 4397   Predcpred 5368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-cnv 4833  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369
This theorem is referenced by: (None)
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