Theorem nffrfor 4347

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 4330	. 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 2319	. . . . . . . 8 |- F/_ x v
4		nffrfor.r	. . . . . . . 8 |- F/_ x R
5		nfcv 2319	. . . . . . . 8 |- F/_ x u
6	3, 4, 5	nfbr 4048	. . . . . . 7 |- F/ x v R u
7		nffrfor.s	. . . . . . . 8 |- F/_ x S
8	7	nfcri 2313	. . . . . . 7 |- F/ x v e. S
9	6, 8	nfim 1572	. . . . . 6 |- F/ x ( v R u -> v e. S )
10	2, 9	nfralxy 2515	. . . . 5 |- F/ x A. v e. A ( v R u -> v e. S )
11	7	nfcri 2313	. . . . 5 |- F/ x u e. S
12	10, 11	nfim 1572	. . . 4 |- F/ x ( A. v e. A ( v R u -> v e. S ) -> u e. S )
13	2, 12	nfralxy 2515	. . 3 |- F/ x A. u e. A ( A. v e. A ( v R u -> v e. S ) -> u e. S )
14	2, 7	nfss 3148	. . 3 |- F/ x A C_ S
15	13, 14	nfim 1572	. 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 1474	1 |- F/ xFrFor R A S

Syntax hints:   -> wi 4   F/wnf 1460   e. wcel 2148   F/_wnfc 2306   A.wral 2455   C_ wss 3129   class class class wbr 4002  FrFor wfrfor 4326

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-frfor 4330

This theorem is referenced by:  nffr  4348