Theorem nfwsuc 36266

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 36260	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		nfwsuc.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
3	2	nfcnv 5868	. . . 4 ⊢ Ⅎ𝑥◡𝑅
4		nfwsuc.2	. . . 4 ⊢ Ⅎ𝑥𝐴
5		nfwsuc.3	. . . 4 ⊢ Ⅎ𝑥𝑋
6	3, 4, 5	nfpred 6311	. . 3 ⊢ Ⅎ𝑥Pred(◡𝑅, 𝐴, 𝑋)
7	6, 4, 2	nfinf 9446	. 2 ⊢ Ⅎ𝑥inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8	1, 7	nfcxfr 2930	1 ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2917  ^◡ccnv 5664  Predcpred 6305  infcinf 9404  wsuccwsuc 36258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5671  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-sup 9405  df-inf 9406  df-wsuc 36260

This theorem is referenced by: (None)