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Theorem predeq1 6178
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq1 (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))

Proof of Theorem predeq1
StepHypRef Expression
1 eqid 2738 . 2 𝐴 = 𝐴
2 eqid 2738 . 2 𝑋 = 𝑋
3 predeq123 6177 . 2 ((𝑅 = 𝑆𝐴 = 𝐴𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
41, 2, 3mp3an23 1455 1 (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  Predcpred 6175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-br 5069  df-opab 5131  df-xp 5572  df-cnv 5574  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176
This theorem is referenced by:  wrecseq123  8069  trpredeq1  9350
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