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Theorem predeq3 28822
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 2467 . 2  |-  R  =  R
2 eqid 2467 . 2  |-  A  =  A
3 predeq123 28819 . 2  |-  ( ( R  =  R  /\  A  =  A  /\  X  =  Y )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y )
)
41, 2, 3mp3an12 1314 1  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   Predcpred 28817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-pred 28818
This theorem is referenced by:  dfpred3g  28829  cbvsetlike  28835  predbrg  28840  preddowncl  28850  wfisg  28863  trpredeq3  28879  trpredlem1  28884  trpredtr  28887  trpredmintr  28888  trpredrec  28895  frmin  28896  frinsg  28899  wfr3g  28916  wfrlem1  28917  wfrlem9  28925  wfrlem14  28930  wfrlem15  28931  wfr2  28934  elwlim  28953  frr3g  28960  frrlem1  28961  frrlem5e  28969
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