Theorem nfwe 5675

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 5654	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffr 5673	. . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
5	2, 3	nfso 5614	. . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴
6	4, 5	nfan 1898	. 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)
7	1, 6	nfxfr 1851	1 ⊢ Ⅎ𝑥 𝑅 We 𝐴

Syntax hints:   ∧ wa 395  Ⅎwnf 1781  Ⅎwnfc 2893   Or wor 5606   Fr wfr 5649   We wwe 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-po 5607  df-so 5608  df-fr 5652  df-we 5654

This theorem is referenced by:  nfoi  9583  aomclem6  43016