Theorem nfwe 4813

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
nffr.r	|- F/_ x R
nffr.a	|- F/_ x A

Assertion

Ref	Expression
nfwe	|- F/ x R We A

Proof of Theorem nfwe

Step	Hyp	Ref	Expression
1		df-we 4798	. 2 |- ( R We A <-> ( R Fr A /\ R Or A ) )
2		nffr.r	. . . 4 |- F/_ x R
3		nffr.a	. . . 4 |- F/_ x A
4	2, 3	nffr 4811	. . 3 |- F/ x R Fr A
5	2, 3	nfso 4764	. . 3 |- F/ x R Or A
6	4, 5	nfan 2013	. 2 |- F/ x ( R Fr A /\ R Or A )
7	1, 6	nfxfr 1698	1 |- F/ x R We A

Syntax hints:   /\ wa 371   F/wnf 1669   F/_wnfc 2581   Or wor 4757   Fr wfr 4793   We wwe 4795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-po 4758  df-so 4759  df-fr 4796  df-we 4798

This theorem is referenced by:  nfoi  8034  aomclem6  35929