Theorem nfwe 5622

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 5602	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffr 5620	. . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
5	2, 3	nfso 5562	. . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴
6	4, 5	nfan 1919	. 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)
7	1, 6	nfxfr 1873	1 ⊢ Ⅎ𝑥 𝑅 We 𝐴

Syntax hints:   ∧ wa 399  Ⅎwnf 1803  Ⅎwnfc 2909   Or wor 5554   Fr wfr 5597   We wwe 5599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-po 5555  df-so 5556  df-fr 5600  df-we 5602

This theorem is referenced by:  nfoi  9462  aomclem6  43633