Theorem nfwsuc 36038

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 36032	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		nfwsuc.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
3	2	nfcnv 5837	. . . 4 ⊢ Ⅎ𝑥◡𝑅
4		nfwsuc.2	. . . 4 ⊢ Ⅎ𝑥𝐴
5		nfwsuc.3	. . . 4 ⊢ Ⅎ𝑥𝑋
6	3, 4, 5	nfpred 6274	. . 3 ⊢ Ⅎ𝑥Pred(◡𝑅, 𝐴, 𝑋)
7	6, 4, 2	nfinf 9400	. 2 ⊢ Ⅎ𝑥inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8	1, 7	nfcxfr 2897	1 ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2884  ^◡ccnv 5633  Predcpred 6268  infcinf 9358  wsuccwsuc 36030

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-sup 9359  df-inf 9360  df-wsuc 36032

This theorem is referenced by: (None)