Theorem nfuni 3829

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 3825	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfuni.1	. . . 4 |- F/_ x A
3		nfv 1538	. . . 4 |- F/ x y e. z
4	2, 3	nfrexxy 2528	. . 3 |- F/ x E. z e. A y e. z
5	4	nfab 2336	. 2 |- F/_ x { y | E. z e. A y e. z }
6	1, 5	nfcxfr 2328	1 |- F/_ x U. A

Syntax hints:   {cab 2174   F/_wnfc 2318   E.wrex 2468   U.cuni 3823

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rex 2473  df-uni 3824

This theorem is referenced by:  nfiota1  5194  nfrecs  6325  nfsup  7008