Theorem nffrfor 4085

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 4068	. 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 2178	. . . . . . . 8 |- F/_ x v
4		nffrfor.r	. . . . . . . 8 |- F/_ x R
5		nfcv 2178	. . . . . . . 8 |- F/_ x u
6	3, 4, 5	nfbr 3808	. . . . . . 7 |- F/ x v R u
7		nffrfor.s	. . . . . . . 8 |- F/_ x S
8	7	nfcri 2172	. . . . . . 7 |- F/ x v e. S
9	6, 8	nfim 1464	. . . . . 6 |- F/ x ( v R u -> v e. S )
10	2, 9	nfralxy 2360	. . . . 5 |- F/ x A. v e. A ( v R u -> v e. S )
11	7	nfcri 2172	. . . . 5 |- F/ x u e. S
12	10, 11	nfim 1464	. . . 4 |- F/ x ( A. v e. A ( v R u -> v e. S ) -> u e. S )
13	2, 12	nfralxy 2360	. . 3 |- F/ x A. u e. A ( A. v e. A ( v R u -> v e. S ) -> u e. S )
14	2, 7	nfss 2938	. . 3 |- F/ x A C_ S
15	13, 14	nfim 1464	. 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 1363	1 |- F/ xFrFor R A S

Syntax hints:   -> wi 4   F/wnf 1349   e. wcel 1393   F/_wnfc 2165   A.wral 2306   C_ wss 2917   class class class wbr 3764  FrFor wfrfor 4064

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-frfor 4068

This theorem is referenced by:  nffr  4086