Theorem nfwe 4357

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 4336	. 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 4351	. . 3 |- F/ x R Fr A
5		nfcv 2319	. . . . . . . . 9 |- F/_ x a
6		nfcv 2319	. . . . . . . . 9 |- F/_ x b
7	5, 2, 6	nfbr 4051	. . . . . . . 8 |- F/ x a R b
8		nfcv 2319	. . . . . . . . 9 |- F/_ x c
9	6, 2, 8	nfbr 4051	. . . . . . . 8 |- F/ x b R c
10	7, 9	nfan 1565	. . . . . . 7 |- F/ x ( a R b /\ b R c )
11	5, 2, 8	nfbr 4051	. . . . . . 7 |- F/ x a R c
12	10, 11	nfim 1572	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
13	3, 12	nfralxy 2515	. . . . 5 |- F/ x A. c e. A ( ( a R b /\ b R c ) -> a R c )
14	3, 13	nfralxy 2515	. . . 4 |- F/ x A. b e. A A. c e. A ( ( a R b /\ b R c ) -> a R c )
15	3, 14	nfralxy 2515	. . 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 1565	. 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 1474	1 |- F/ x R We A

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1460   F/_wnfc 2306   A.wral 2455   class class class wbr 4005   Fr wfr 4330   We wwe 4332

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 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-frfor 4333  df-frind 4334  df-wetr 4336

This theorem is referenced by: (None)