Theorem nffrfor 4326

Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nffrfor.r	|- F/_ x R
nffrfor.a	|- F/_ x A
nffrfor.s	|- F/_ x S

Assertion

Ref	Expression
nffrfor	|- F/ xFrFor R A S

Proof of Theorem nffrfor

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frfor 4309	. 2 |- (FrFor R A S <-> ( A. u e. A ( A. v e. A ( v R u -> v e. S ) -> u e. S ) -> A C_ S ) )
2		nffrfor.a	. . . 4 |- F/_ x A
3		nfcv 2308	. . . . . . . 8 |- F/_ x v
4		nffrfor.r	. . . . . . . 8 |- F/_ x R
5		nfcv 2308	. . . . . . . 8 |- F/_ x u
6	3, 4, 5	nfbr 4028	. . . . . . 7 |- F/ x v R u
7		nffrfor.s	. . . . . . . 8 |- F/_ x S
8	7	nfcri 2302	. . . . . . 7 |- F/ x v e. S
9	6, 8	nfim 1560	. . . . . 6 |- F/ x ( v R u -> v e. S )
10	2, 9	nfralxy 2504	. . . . 5 |- F/ x A. v e. A ( v R u -> v e. S )
11	7	nfcri 2302	. . . . 5 |- F/ x u e. S
12	10, 11	nfim 1560	. . . 4 |- F/ x ( A. v e. A ( v R u -> v e. S ) -> u e. S )
13	2, 12	nfralxy 2504	. . 3 |- F/ x A. u e. A ( A. v e. A ( v R u -> v e. S ) -> u e. S )
14	2, 7	nfss 3135	. . 3 |- F/ x A C_ S
15	13, 14	nfim 1560	. 2 |- F/ x ( A. u e. A ( A. v e. A ( v R u -> v e. S ) -> u e. S ) -> A C_ S )
16	1, 15	nfxfr 1462	1 |- F/ xFrFor R A S

Syntax hints:   -> wi 4   F/wnf 1448   e. wcel 2136   F/_wnfc 2295   A.wral 2444   C_ wss 3116   class class class wbr 3982  FrFor wfrfor 4305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-frfor 4309

This theorem is referenced by:  nffr  4327