Theorem nfwe 4827

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 4812	. 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 4825	. . 3 |- F/ x R Fr A
5	2, 3	nfso 4778	. . 3 |- F/ x R Or A
6	4, 5	nfan 1985	. 2 |- F/ x ( R Fr A /\ R Or A )
7	1, 6	nfxfr 1693	1 |- F/ x R We A

Syntax hints:   /\ wa 371   F/wnf 1664   F/_wnfc 2571   Or wor 4771   Fr wfr 4807   We wwe 4809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-po 4772  df-so 4773  df-fr 4810  df-we 4812

This theorem is referenced by:  nfoi  8033  aomclem6  35881