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Theorem predeq2 6133
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 2739 . 2 𝑅 = 𝑅
2 eqid 2739 . 2 𝑋 = 𝑋
3 predeq123 6131 . 2 ((𝑅 = 𝑅𝐴 = 𝐵𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
41, 2, 3mp3an13 1453 1 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Predcpred 6129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3401  df-un 3849  df-in 3851  df-ss 3861  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-xp 5532  df-cnv 5534  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130
This theorem is referenced by:  wrecseq123  7980  wfrlem5  7991  prednn  13124  prednn0  13125  trpredeq2  33368  frmin  33395  fprlem1  33460  frrlem15  33465
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