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Theorem predeq3 5587
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 2609 . 2 𝑅 = 𝑅
2 eqid 2609 . 2 𝐴 = 𝐴
3 predeq123 5584 . 2 ((𝑅 = 𝑅𝐴 = 𝐴𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
41, 2, 3mp3an12 1405 1 (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  Predcpred 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583
This theorem is referenced by:  dfpred3g  5594  predbrg  5603  preddowncl  5610  wfisg  5618  wfr3g  7278  wfrlem1  7279  wfrdmcl  7288  wfrlem14  7293  wfrlem15  7294  wfrlem17  7296  wfr2a  7297  trpredeq3  30800  trpredlem1  30805  trpredtr  30808  trpredmintr  30809  trpredrec  30816  frmin  30817  frinsg  30820  elwlim  30847  elwlimOLD  30848  frr3g  30857  frrlem1  30858  frrlem5e  30866  csbwrecsg  32173
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