Theorem nfun 3319

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 3161	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2333	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2333	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfor 1588	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	6	nfab 2344	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
8	1, 7	nfcxfr 2336	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 709   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326   ∪ cun 3155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-un 3161

This theorem is referenced by:  nfsuc  4443  nfdju  7108