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Theorem predeq3 5827
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 2771 . 2 𝑅 = 𝑅
2 eqid 2771 . 2 𝐴 = 𝐴
3 predeq123 5824 . 2 ((𝑅 = 𝑅𝐴 = 𝐴𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
41, 2, 3mp3an12 1562 1 (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  Predcpred 5822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823
This theorem is referenced by:  dfpred3g  5834  predbrg  5843  preddowncl  5850  wfisg  5858  wfr3g  7565  wfrlem1  7566  wfrdmcl  7576  wfrlem14  7581  wfrlem15  7582  wfrlem17  7584  wfr2a  7585  trpredeq3  32058  trpredlem1  32063  trpredtr  32066  trpredmintr  32067  trpredrec  32074  frpoinsg  32078  frmin  32079  frinsg  32082  elwlim  32105  frr3g  32116  frrlem1  32117  frrlem5e  32125  csbwrecsg  33510
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