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Theorem nfpred 6326
Description: Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)
Hypotheses
Ref Expression
nfpred.1 𝑥𝑅
nfpred.2 𝑥𝐴
nfpred.3 𝑥𝑋
Assertion
Ref Expression
nfpred 𝑥Pred(𝑅, 𝐴, 𝑋)

Proof of Theorem nfpred
StepHypRef Expression
1 df-pred 6321 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 nfpred.2 . . 3 𝑥𝐴
3 nfpred.1 . . . . 5 𝑥𝑅
43nfcnv 5889 . . . 4 𝑥𝑅
5 nfpred.3 . . . . 5 𝑥𝑋
65nfsn 4707 . . . 4 𝑥{𝑋}
74, 6nfima 6086 . . 3 𝑥(𝑅 “ {𝑋})
82, 7nfin 4224 . 2 𝑥(𝐴 ∩ (𝑅 “ {𝑋}))
91, 8nfcxfr 2903 1 𝑥Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2890  cin 3950  {csn 4626  ccnv 5684  cima 5688  Predcpred 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321
This theorem is referenced by:  nffrecs  8308  nfwrecsOLD  8342  nfwsuc  35819  nfwlim  35823
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