Theorem nffrfor 4445

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 4428	. 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 2374	. . . . . . . 8 |- F/_ x v
4		nffrfor.r	. . . . . . . 8 |- F/_ x R
5		nfcv 2374	. . . . . . . 8 |- F/_ x u
6	3, 4, 5	nfbr 4135	. . . . . . 7 |- F/ x v R u
7		nffrfor.s	. . . . . . . 8 |- F/_ x S
8	7	nfcri 2368	. . . . . . 7 |- F/ x v e. S
9	6, 8	nfim 1620	. . . . . 6 |- F/ x ( v R u -> v e. S )
10	2, 9	nfralxy 2570	. . . . 5 |- F/ x A. v e. A ( v R u -> v e. S )
11	7	nfcri 2368	. . . . 5 |- F/ x u e. S
12	10, 11	nfim 1620	. . . 4 |- F/ x ( A. v e. A ( v R u -> v e. S ) -> u e. S )
13	2, 12	nfralxy 2570	. . 3 |- F/ x A. u e. A ( A. v e. A ( v R u -> v e. S ) -> u e. S )
14	2, 7	nfss 3220	. . 3 |- F/ x A C_ S
15	13, 14	nfim 1620	. 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 1522	1 |- F/ xFrFor R A S

Syntax hints:   -> wi 4   F/wnf 1508   e. wcel 2202   F/_wnfc 2361   A.wral 2510   C_ wss 3200   class class class wbr 4088  FrFor wfrfor 4424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-frfor 4428

This theorem is referenced by:  nffr  4446