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Theorem nfwsuc 31008
 Description: Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
Hypotheses
Ref Expression
nfwsuc.1 𝑥𝑅
nfwsuc.2 𝑥𝐴
nfwsuc.3 𝑥𝑋
Assertion
Ref Expression
nfwsuc 𝑥wsuc(𝑅, 𝐴, 𝑋)

Proof of Theorem nfwsuc
StepHypRef Expression
1 df-wsuc 31000 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 nfwsuc.1 . . . . 5 𝑥𝑅
32nfcnv 5223 . . . 4 𝑥𝑅
4 nfwsuc.2 . . . 4 𝑥𝐴
5 nfwsuc.3 . . . 4 𝑥𝑋
63, 4, 5nfpred 5602 . . 3 𝑥Pred(𝑅, 𝐴, 𝑋)
76, 4, 2nfinf 8271 . 2 𝑥inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
81, 7nfcxfr 2749 1 𝑥wsuc(𝑅, 𝐴, 𝑋)
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnfc 2738  ◡ccnv 5037  Predcpred 5596  infcinf 8230  wsuccwsuc 30996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-sup 8231  df-inf 8232  df-wsuc 31000 This theorem is referenced by: (None)
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