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Theorem predeq3 25435
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 2435 . 2  |-  R  =  R
2 eqid 2435 . 2  |-  A  =  A
3 predeq123 25432 . 2  |-  ( ( R  =  R  /\  A  =  A  /\  X  =  Y )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y )
)
41, 2, 3mp3an12 1269 1  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   Predcpred 25430
This theorem is referenced by:  dfpred3g  25442  cbvsetlike  25448  predbrg  25453  preddowncl  25463  wfisg  25476  trpredeq3  25492  trpredlem1  25497  trpredtr  25500  trpredmintr  25501  trpredrec  25508  frmin  25509  frinsg  25512  wfr3g  25529  wfrlem1  25530  wfrlem9  25538  wfrlem14  25543  wfrlem15  25544  wfr2  25547  elwlim  25566  frr3g  25573  frrlem1  25574  frrlem5e  25582
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  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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