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Theorem predeq3 6258
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 2733 . 2 𝑅 = 𝑅
2 eqid 2733 . 2 𝐴 = 𝐴
3 predeq123 6255 . 2 ((𝑅 = 𝑅𝐴 = 𝐴𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
41, 2, 3mp3an12 1452 1 (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Predcpred 6253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254
This theorem is referenced by:  dfpred3g  6266  predbrg  6276  preddowncl  6287  frpoinsg  6298  wfisgOLD  6309  frpoins3xpg  8073  frpoins3xp3g  8074  xpord2pred  8078  sexp2  8079  xpord3pred  8085  sexp3  8086  csbfrecsg  8216  fpr3g  8217  frrlem1  8218  frrlem12  8229  frrlem13  8230  fpr2a  8234  frrdmcl  8240  fprresex  8242  wfr3g  8254  wfrlem1OLD  8255  wfrdmclOLD  8264  wfrlem14OLD  8269  wfrlem15OLD  8270  wfrlem17OLD  8272  wfr2aOLD  8273  ttrclselem1  9666  ttrclselem2  9667  frmin  9690  frinsg  9692  frr3g  9697  frr2  9701  elwlim  34454
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