Theorem nfwe 4769

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 4754	. 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 4767	. . 3 |- F/ x R Fr A
5	2, 3	nfso 4720	. . 3 |- F/ x R Or A
6	4, 5	nfan 1936	. 2 |- F/ x ( R Fr A /\ R Or A )
7	1, 6	nfxfr 1653	1 |- F/ x R We A

Syntax hints:   /\ wa 367   F/wnf 1624   F/_wnfc 2530   Or wor 4713   Fr wfr 4749   We wwe 4751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-po 4714  df-so 4715  df-fr 4752  df-we 4754

This theorem is referenced by:  nfoi  7854  aomclem6  31171