Theorem nfwsuc 35799

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 35793	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		nfwsuc.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
3	2	nfcnv 5832	. . . 4 ⊢ Ⅎ𝑥◡𝑅
4		nfwsuc.2	. . . 4 ⊢ Ⅎ𝑥𝐴
5		nfwsuc.3	. . . 4 ⊢ Ⅎ𝑥𝑋
6	3, 4, 5	nfpred 6267	. . 3 ⊢ Ⅎ𝑥Pred(◡𝑅, 𝐴, 𝑋)
7	6, 4, 2	nfinf 9410	. 2 ⊢ Ⅎ𝑥inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8	1, 7	nfcxfr 2889	1 ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2876  ^◡ccnv 5630  Predcpred 6261  infcinf 9368  wsuccwsuc 35791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-sup 9369  df-inf 9370  df-wsuc 35793

This theorem is referenced by: (None)