Theorem nfwsuc 33739

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	⊢ Ⅎ𝑥𝑅
nfwsuc.2	⊢ Ⅎ𝑥𝐴
nfwsuc.3	⊢ Ⅎ𝑥𝑋

Assertion

Ref	Expression
nfwsuc	⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)

Proof of Theorem nfwsuc

Step	Hyp	Ref	Expression
1		df-wsuc 33733	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		nfwsuc.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
3	2	nfcnv 5776	. . . 4 ⊢ Ⅎ𝑥◡𝑅
4		nfwsuc.2	. . . 4 ⊢ Ⅎ𝑥𝐴
5		nfwsuc.3	. . . 4 ⊢ Ⅎ𝑥𝑋
6	3, 4, 5	nfpred 6196	. . 3 ⊢ Ⅎ𝑥Pred(◡𝑅, 𝐴, 𝑋)
7	6, 4, 2	nfinf 9171	. 2 ⊢ Ⅎ𝑥inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8	1, 7	nfcxfr 2904	1 ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2886  ^◡ccnv 5579  Predcpred 6190  infcinf 9130  wsuccwsuc 33731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-sup 9131  df-inf 9132  df-wsuc 33733

This theorem is referenced by: (None)