Theorem nfuni 3893

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 3889	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfuni.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1574	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfrexw 2569	. . 3 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2377	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2369	1 ⊢ Ⅎ𝑥∪ 𝐴

Syntax hints:  {cab 2215  Ⅎwnfc 2359  ∃wrex 2509  ∪ cuni 3887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3888

This theorem is referenced by:  nfiota1  5279  iotaexab  5296  nfrecs  6451  nfsup  7155