Theorem nfun 4111

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 4094	. . 3 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2891	. . . 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 2887	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 848   ∈ wcel 2114  Ⅎwnfc 2884   ∪ cun 3888

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-v 3432  df-un 3895

This theorem is referenced by:  nfsymdif  4198  csbun  4382  iunxdif3  5038  nfsuc  6392  nfsup  9358  nfdju  9825  iunconn  23406  nosupbnd2  27697  noinfbnd2  27712  ordtconnlem1  34087  esumsplit  34216  measvuni  34377  bnj958  35101  bnj1000  35102  bnj1408  35197  bnj1446  35206  bnj1447  35207  bnj1448  35208  bnj1466  35214  bnj1467  35215  rdgssun  37711  exrecfnlem  37712  poimirlem16  37974  poimirlem19  37977  pimxrneun  45937  pimrecltpos  47157