Theorem nfwe 4191

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

Hypotheses

Ref	Expression
nfwe.r	|- F/_ x R
nfwe.a	|- F/_ x A

Assertion

Ref	Expression
nfwe	|- F/ x R We A

Proof of Theorem nfwe

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wetr 4170	. 2 |- ( R We A <-> ( R Fr A /\ A. a e. A A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c ) ) )
2		nfwe.r	. . . 4 |- F/_ x R
3		nfwe.a	. . . 4 |- F/_ x A
4	2, 3	nffr 4185	. . 3 |- F/ x R Fr A
5		nfcv 2229	. . . . . . . . 9 |- F/_ x a
6		nfcv 2229	. . . . . . . . 9 |- F/_ x b
7	5, 2, 6	nfbr 3895	. . . . . . . 8 |- F/ x a R b
8		nfcv 2229	. . . . . . . . 9 |- F/_ x c
9	6, 2, 8	nfbr 3895	. . . . . . . 8 |- F/ x b R c
10	7, 9	nfan 1503	. . . . . . 7 |- F/ x ( a R b /\ b R c )
11	5, 2, 8	nfbr 3895	. . . . . . 7 |- F/ x a R c
12	10, 11	nfim 1510	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
13	3, 12	nfralxy 2415	. . . . 5 |- F/ x A. c e. A ( ( a R b /\ b R c ) -> a R c )
14	3, 13	nfralxy 2415	. . . 4 |- F/ x A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c )
15	3, 14	nfralxy 2415	. . 3 |- F/ x A. a e. A A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c )
16	4, 15	nfan 1503	. 2 |- F/ x ( R Fr A /\ A. a e. A A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c ) )
17	1, 16	nfxfr 1409	1 |- F/ x R We A

Syntax hints:   -> wi 4   /\ wa 103   F/wnf 1395   F/_wnfc 2216   A.wral 2360   class class class wbr 3851   Fr wfr 4164   We wwe 4166

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-frfor 4167  df-frind 4168  df-wetr 4170

This theorem is referenced by: (None)