Theorem nfuni 3789

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	|- F/_ x A

Assertion

Ref	Expression
nfuni	|- F/_ x U. A

Proof of Theorem nfuni

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

Step	Hyp	Ref	Expression
1		dfuni2 3785	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfuni.1	. . . 4 |- F/_ x A
3		nfv 1515	. . . 4 |- F/ x y e. z
4	2, 3	nfrexxy 2503	. . 3 |- F/ x E. z e. A y e. z
5	4	nfab 2311	. 2 |- F/_ x { y | E. z e. A y e. z }
6	1, 5	nfcxfr 2303	1 |- F/_ x U. A

Syntax hints:   {cab 2150   F/_wnfc 2293   E.wrex 2443   U.cuni 3783

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-uni 3784

This theorem is referenced by:  nfiota1  5149  nfrecs  6266  nfsup  6948