Theorem nfwsuc 31764

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	|- F/_ x R
nfwsuc.2	|- F/_ x A
nfwsuc.3	|- F/_ x X

Assertion

Ref	Expression
nfwsuc	|- F/_ xwsuc ( R , A , X )

Proof of Theorem nfwsuc

Step	Hyp	Ref	Expression
1		df-wsuc 31756	. 2 |- wsuc ( R , A , X ) = inf ( Pred ( `' R , A , X ) , A , R )
2		nfwsuc.1	. . . . 5 |- F/_ x R
3	2	nfcnv 5301	. . . 4 |- F/_ x `' R
4		nfwsuc.2	. . . 4 |- F/_ x A
5		nfwsuc.3	. . . 4 |- F/_ x X
6	3, 4, 5	nfpred 5685	. . 3 |- F/_ x Pred ( `' R , A , X )
7	6, 4, 2	nfinf 8388	. 2 |- F/_ xinf ( Pred ( `' R , A , X ) , A , R )
8	1, 7	nfcxfr 2762	1 |- F/_ xwsuc ( R , A , X )

Syntax hints:   F/_wnfc 2751   `'ccnv 5113   Predcpred 5679  infcinf 8347  wsuccwsuc 31752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-sup 8348  df-inf 8349  df-wsuc 31756

This theorem is referenced by: (None)