Theorem nfun 4110

Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2147, ax-11 2163, ax-12 2185. (Revised by SN, 14-May-2025.)

Hypotheses

Ref	Expression
nfun.1	⊢ Ⅎ𝑥𝐴
nfun.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfun	⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Proof of Theorem nfun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 4093	. . 3 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2890	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2890	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfor 1906	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1855	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∪ 𝐵)
8	7	nfci 2886	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 848   ∈ wcel 2114  Ⅎwnfc 2883   ∪ cun 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-v 3431  df-un 3894

This theorem is referenced by:  nfsymdif  4197  csbun  4381  iunxdif3  5037  nfsuc  6397  nfsup  9364  nfdju  9831  iunconn  23393  nosupbnd2  27680  noinfbnd2  27695  ordtconnlem1  34068  esumsplit  34197  measvuni  34358  bnj958  35082  bnj1000  35083  bnj1408  35178  bnj1446  35187  bnj1447  35188  bnj1448  35189  bnj1466  35195  bnj1467  35196  rdgssun  37694  exrecfnlem  37695  poimirlem16  37957  poimirlem19  37960  pimxrneun  45916  pimrecltpos  47136