Theorem nfwe 5419

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	⊢ Ⅎ𝑥𝑅
nffr.a	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfwe	⊢ Ⅎ𝑥 𝑅 We 𝐴

Proof of Theorem nfwe

Step	Hyp	Ref	Expression
1		df-we 5404	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffr 5417	. . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
5	2, 3	nfso 5368	. . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴
6	4, 5	nfan 1881	. 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)
7	1, 6	nfxfr 1834	1 ⊢ Ⅎ𝑥 𝑅 We 𝐴

Syntax hints:   ∧ wa 396  Ⅎwnf 1765  Ⅎwnfc 2933   Or wor 5361   Fr wfr 5399   We wwe 5401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-po 5362  df-so 5363  df-fr 5402  df-we 5404

This theorem is referenced by:  nfoi  8824  aomclem6  39144