Theorem nffrfor 4265

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 4248	. 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 2279	. . . . . . . 8 |- F/_ x v
4		nffrfor.r	. . . . . . . 8 |- F/_ x R
5		nfcv 2279	. . . . . . . 8 |- F/_ x u
6	3, 4, 5	nfbr 3969	. . . . . . 7 |- F/ x v R u
7		nffrfor.s	. . . . . . . 8 |- F/_ x S
8	7	nfcri 2273	. . . . . . 7 |- F/ x v e. S
9	6, 8	nfim 1551	. . . . . 6 |- F/ x ( v R u -> v e. S )
10	2, 9	nfralxy 2469	. . . . 5 |- F/ x A. v e. A ( v R u -> v e. S )
11	7	nfcri 2273	. . . . 5 |- F/ x u e. S
12	10, 11	nfim 1551	. . . 4 |- F/ x ( A. v e. A ( v R u -> v e. S ) -> u e. S )
13	2, 12	nfralxy 2469	. . 3 |- F/ x A. u e. A ( A. v e. A ( v R u -> v e. S ) -> u e. S )
14	2, 7	nfss 3085	. . 3 |- F/ x A C_ S
15	13, 14	nfim 1551	. 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 1450	1 |- F/ xFrFor R A S

Syntax hints:   -> wi 4   F/wnf 1436   e. wcel 1480   F/_wnfc 2266   A.wral 2414   C_ wss 3066   class class class wbr 3924  FrFor wfrfor 4244

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-frfor 4248

This theorem is referenced by:  nffr  4266