Theorem nfwe 4420

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 4399	. 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 4414	. . 3 |- F/ x R Fr A
5		nfcv 2350	. . . . . . . . 9 |- F/_ x a
6		nfcv 2350	. . . . . . . . 9 |- F/_ x b
7	5, 2, 6	nfbr 4106	. . . . . . . 8 |- F/ x a R b
8		nfcv 2350	. . . . . . . . 9 |- F/_ x c
9	6, 2, 8	nfbr 4106	. . . . . . . 8 |- F/ x b R c
10	7, 9	nfan 1589	. . . . . . 7 |- F/ x ( a R b /\ b R c )
11	5, 2, 8	nfbr 4106	. . . . . . 7 |- F/ x a R c
12	10, 11	nfim 1596	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
13	3, 12	nfralxy 2546	. . . . 5 |- F/ x A. c e. A ( ( a R b /\ b R c ) -> a R c )
14	3, 13	nfralxy 2546	. . . 4 |- F/ x A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c )
15	3, 14	nfralxy 2546	. . 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 1589	. 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 1498	1 |- F/ x R We A

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1484   F/_wnfc 2337   A.wral 2486   class class class wbr 4059   Fr wfr 4393   We wwe 4395

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-frfor 4396  df-frind 4397  df-wetr 4399

This theorem is referenced by: (None)