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Theorem predeq3 5601
 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 2610 . 2 𝑅 = 𝑅
2 eqid 2610 . 2 𝐴 = 𝐴
3 predeq123 5598 . 2 ((𝑅 = 𝑅𝐴 = 𝐴𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
41, 2, 3mp3an12 1406 1 (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Predcpred 5596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597 This theorem is referenced by:  dfpred3g  5608  predbrg  5617  preddowncl  5624  wfisg  5632  wfr3g  7300  wfrlem1  7301  wfrdmcl  7310  wfrlem14  7315  wfrlem15  7316  wfrlem17  7318  wfr2a  7319  trpredeq3  30966  trpredlem1  30971  trpredtr  30974  trpredmintr  30975  trpredrec  30982  frmin  30983  frinsg  30986  elwlim  31013  elwlimOLD  31014  frr3g  31023  frrlem1  31024  frrlem5e  31032  csbwrecsg  32349
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