Theorem nfwe 4815

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 4800	. 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 4813	. . 3 |- F/ x R Fr A
5	2, 3	nfso 4766	. . 3 |- F/ x R Or A
6	4, 5	nfan 2031	. 2 |- F/ x ( R Fr A /\ R Or A )
7	1, 6	nfxfr 1704	1 |- F/ x R We A

Syntax hints:   /\ wa 376   F/wnf 1675   F/_wnfc 2599   Or wor 4759   Fr wfr 4795   We wwe 4797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-po 4760  df-so 4761  df-fr 4798  df-we 4800

This theorem is referenced by:  nfoi  8047  aomclem6  35988