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Theorem nfpred 6131
 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 6126 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 nfpred.2 . . 3 𝑥𝐴
3 nfpred.1 . . . . 5 𝑥𝑅
43nfcnv 5718 . . . 4 𝑥𝑅
5 nfpred.3 . . . . 5 𝑥𝑋
65nfsn 4600 . . . 4 𝑥{𝑋}
74, 6nfima 5909 . . 3 𝑥(𝑅 “ {𝑋})
82, 7nfin 4121 . 2 𝑥(𝐴 ∩ (𝑅 “ {𝑋}))
91, 8nfcxfr 2917 1 𝑥Pred(𝑅, 𝐴, 𝑋)
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnfc 2899   ∩ cin 3857  {csn 4522  ◡ccnv 5523   “ cima 5527  Predcpred 6125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126 This theorem is referenced by:  nfwrecs  7959  nfwsuc  33366  nfwlim  33370  nffrecs  33381
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