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Theorem nfpred 5905
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 5900 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 nfpred.2 . . 3 𝑥𝐴
3 nfpred.1 . . . . 5 𝑥𝑅
43nfcnv 5509 . . . 4 𝑥𝑅
5 nfpred.3 . . . . 5 𝑥𝑋
65nfsn 4441 . . . 4 𝑥{𝑋}
74, 6nfima 5691 . . 3 𝑥(𝑅 “ {𝑋})
82, 7nfin 4024 . 2 𝑥(𝐴 ∩ (𝑅 “ {𝑋}))
91, 8nfcxfr 2953 1 𝑥Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2942  cin 3775  {csn 4377  ccnv 5317  cima 5321  Predcpred 5899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-xp 5324  df-cnv 5326  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900
This theorem is referenced by:  nfwrecs  7647  nfwsuc  32089  nfwlim  32093  nffrecs  32104
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