Theorem nfwe 4110

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 4089	. 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 4104	. . 3 |- F/ x R Fr A
5		nfcv 2219	. . . . . . . . 9 |- F/_ x a
6		nfcv 2219	. . . . . . . . 9 |- F/_ x b
7	5, 2, 6	nfbr 3829	. . . . . . . 8 |- F/ x a R b
8		nfcv 2219	. . . . . . . . 9 |- F/_ x c
9	6, 2, 8	nfbr 3829	. . . . . . . 8 |- F/ x b R c
10	7, 9	nfan 1497	. . . . . . 7 |- F/ x ( a R b /\ b R c )
11	5, 2, 8	nfbr 3829	. . . . . . 7 |- F/ x a R c
12	10, 11	nfim 1504	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
13	3, 12	nfralxy 2402	. . . . 5 |- F/ x A. c e. A ( ( a R b /\ b R c ) -> a R c )
14	3, 13	nfralxy 2402	. . . 4 |- F/ x A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c )
15	3, 14	nfralxy 2402	. . 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 1497	. 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 1403	1 |- F/ x R We A

Syntax hints:   -> wi 4   /\ wa 102   F/wnf 1389   F/_wnfc 2206   A.wral 2348   class class class wbr 3785   Fr wfr 4083   We wwe 4085

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-frfor 4086  df-frind 4087  df-wetr 4089

This theorem is referenced by: (None)