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Theorem predeq3 5370
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq3  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )

Proof of Theorem predeq3
StepHypRef Expression
1 eqid 2402 . 2  |-  R  =  R
2 eqid 2402 . 2  |-  A  =  A
3 predeq123 5367 . 2  |-  ( ( R  =  R  /\  A  =  A  /\  X  =  Y )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y )
)
41, 2, 3mp3an12 1316 1  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   Predcpred 5365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366
This theorem is referenced by:  dfpred3g  5377  predbrg  5386  preddowncl  5393  wfisg  5401  wfr3g  7018  wfrlem1  7019  wfrdmcl  7028  wfrlem14  7033  wfrlem15  7034  wfrlem17  7036  wfr2a  7037  trpredeq3  30023  trpredlem1  30028  trpredtr  30031  trpredmintr  30032  trpredrec  30039  frmin  30040  frinsg  30043  elwlim  30066  frr3g  30073  frrlem1  30074  frrlem5e  30082
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