Theorem nfun 3157

Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

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		df-un 3004	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2223	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2223	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfor 1512	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	6	nfab 2234	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
8	1, 7	nfcxfr 2226	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 665   ∈ wcel 1439  {cab 2075  Ⅎwnfc 2216   ∪ cun 2998

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-un 3004

This theorem is referenced by:  nfsuc  4244  nfdju  6789