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Theorem nfpred 6296
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 6291 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 nfpred.2 . . 3 𝑥𝐴
3 nfpred.1 . . . . 5 𝑥𝑅
43nfcnv 5869 . . . 4 𝑥𝑅
5 nfpred.3 . . . . 5 𝑥𝑋
65nfsn 4704 . . . 4 𝑥{𝑋}
74, 6nfima 6058 . . 3 𝑥(𝑅 “ {𝑋})
82, 7nfin 4209 . 2 𝑥(𝐴 ∩ (𝑅 “ {𝑋}))
91, 8nfcxfr 2893 1 𝑥Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2875  cin 3940  {csn 4621  ccnv 5666  cima 5670  Predcpred 6290
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291
This theorem is referenced by:  nffrecs  8264  nfwrecsOLD  8298  nfwsuc  35313  nfwlim  35317
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