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Theorem nfpred 6310
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 6305 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 nfpred.2 . . 3 𝑥𝐴
3 nfpred.1 . . . . 5 𝑥𝑅
43nfcnv 5881 . . . 4 𝑥𝑅
5 nfpred.3 . . . . 5 𝑥𝑋
65nfsn 4712 . . . 4 𝑥{𝑋}
74, 6nfima 6071 . . 3 𝑥(𝑅 “ {𝑋})
82, 7nfin 4216 . 2 𝑥(𝐴 ∩ (𝑅 “ {𝑋}))
91, 8nfcxfr 2897 1 𝑥Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2879  cin 3946  {csn 4629  ccnv 5677  cima 5681  Predcpred 6304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305
This theorem is referenced by:  nffrecs  8289  nfwrecsOLD  8323  nfwsuc  35414  nfwlim  35418
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