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Theorem predeq2 5899
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 2797 . 2 𝑅 = 𝑅
2 eqid 2797 . 2 𝑋 = 𝑋
3 predeq123 5897 . 2 ((𝑅 = 𝑅𝐴 = 𝐵𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
41, 2, 3mp3an13 1577 1 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  Predcpred 5895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896
This theorem is referenced by:  wrecseq123  7644  wfrlem5  7656  prednn  12713  prednn0  12714  trpredeq2  32224  frmin  32246  frrlem5  32288
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