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Theorem predeq2 5586
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq2 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predeq2
StepHypRef Expression
1 eqid 2609 . 2 𝑅 = 𝑅
2 eqid 2609 . 2 𝑋 = 𝑋
3 predeq123 5584 . 2 ((𝑅 = 𝑅𝐴 = 𝐵𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
41, 2, 3mp3an13 1406 1 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
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:  wrecseq123  7273  wfrlem5  7284  prednn  12289  prednn0  12290  trpredeq2  30799  frmin  30817  frrlem5  30862
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