Theorem nfwe 4481

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 4460	. 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 4475	. . 3 |- F/ x R Fr A
5		nfcv 2386	. . . . . . . . 9 |- F/_ x a
6		nfcv 2386	. . . . . . . . 9 |- F/_ x b
7	5, 2, 6	nfbr 4161	. . . . . . . 8 |- F/ x a R b
8		nfcv 2386	. . . . . . . . 9 |- F/_ x c
9	6, 2, 8	nfbr 4161	. . . . . . . 8 |- F/ x b R c
10	7, 9	nfan 1614	. . . . . . 7 |- F/ x ( a R b /\ b R c )
11	5, 2, 8	nfbr 4161	. . . . . . 7 |- F/ x a R c
12	10, 11	nfim 1621	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
13	3, 12	nfralxy 2582	. . . . 5 |- F/ x A. c e. A ( ( a R b /\ b R c ) -> a R c )
14	3, 13	nfralxy 2582	. . . 4 |- F/ x A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c )
15	3, 14	nfralxy 2582	. . 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 1614	. 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 1523	1 |- F/ x R We A

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1509   F/_wnfc 2373   A.wral 2522   class class class wbr 4114   Fr wfr 4454   We wwe 4456

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-frfor 4457  df-frind 4458  df-wetr 4460

This theorem is referenced by: (None)