Theorem nfun 4122

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 2146, ax-11 2162, ax-12 2184. (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 4105	. . 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 1905	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1854	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∪ 𝐵)
8	7	nfci 2886	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 847   ∈ wcel 2113  Ⅎwnfc 2883   ∪ cun 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-v 3442  df-un 3906

This theorem is referenced by:  nfsymdif  4209  csbun  4393  iunxdif3  5050  nfsuc  6391  nfsup  9354  nfdju  9819  iunconn  23372  nosupbnd2  27684  noinfbnd2  27699  ordtconnlem1  34081  esumsplit  34210  measvuni  34371  bnj958  35096  bnj1000  35097  bnj1408  35192  bnj1446  35201  bnj1447  35202  bnj1448  35203  bnj1466  35209  bnj1467  35210  rdgssun  37583  exrecfnlem  37584  poimirlem16  37837  poimirlem19  37840  pimxrneun  45732  pimrecltpos  46952