Theorem nfuni 3795

Description: Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
nfuni.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfuni	⊢ Ⅎ𝑥∪ 𝐴

Proof of Theorem nfuni

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfuni2 3791	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfuni.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1516	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfrexxy 2505	. . 3 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2313	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2305	1 ⊢ Ⅎ𝑥∪ 𝐴

Syntax hints:  {cab 2151  Ⅎwnfc 2295  ∃wrex 2445  ∪ cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790

This theorem is referenced by:  nfiota1  5155  nfrecs  6275  nfsup  6957