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Theorem predeq2 6148
 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 2825 . 2 𝑅 = 𝑅
2 eqid 2825 . 2 𝑋 = 𝑋
3 predeq123 6146 . 2 ((𝑅 = 𝑅𝐴 = 𝐵𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
41, 2, 3mp3an13 1445 1 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530  Predcpred 6144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145 This theorem is referenced by:  wrecseq123  7942  wfrlem5  7953  prednn  13023  prednn0  13024  trpredeq2  32947  frmin  32971  fprlem1  33024  frrlem15  33029
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