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Theorem nfwsuc 35800
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 35794 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 nfwsuc.1 . . . . 5 𝑥𝑅
32nfcnv 5892 . . . 4 𝑥𝑅
4 nfwsuc.2 . . . 4 𝑥𝐴
5 nfwsuc.3 . . . 4 𝑥𝑋
63, 4, 5nfpred 6328 . . 3 𝑥Pred(𝑅, 𝐴, 𝑋)
76, 4, 2nfinf 9520 . 2 𝑥inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
81, 7nfcxfr 2901 1 𝑥wsuc(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2888  ccnv 5688  Predcpred 6322  infcinf 9479  wsuccwsuc 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-sup 9480  df-inf 9481  df-wsuc 35794
This theorem is referenced by: (None)
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