Theorem nfwsuc 35545

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 35539	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		nfwsuc.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
3	2	nfcnv 5881	. . . 4 ⊢ Ⅎ𝑥◡𝑅
4		nfwsuc.2	. . . 4 ⊢ Ⅎ𝑥𝐴
5		nfwsuc.3	. . . 4 ⊢ Ⅎ𝑥𝑋
6	3, 4, 5	nfpred 6312	. . 3 ⊢ Ⅎ𝑥Pred(◡𝑅, 𝐴, 𝑋)
7	6, 4, 2	nfinf 9507	. 2 ⊢ Ⅎ𝑥inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8	1, 7	nfcxfr 2889	1 ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2875  ^◡ccnv 5677  Predcpred 6306  infcinf 9466  wsuccwsuc 35537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-sup 9467  df-inf 9468  df-wsuc 35539

This theorem is referenced by: (None)