Theorem nfuni 3925

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 3921	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfuni.1	. . . 4 |- F/_ x A
3		nfv 1577	. . . 4 |- F/ x y e. z
4	2, 3	nfrexw 2583	. . 3 |- F/ x E. z e. A y e. z
5	4	nfab 2391	. 2 |- F/_ x { y | E. z e. A y e. z }
6	1, 5	nfcxfr 2383	1 |- F/_ x U. A

Syntax hints:   {cab 2220   F/_wnfc 2373   E.wrex 2523   U.cuni 3919

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-uni 3920

This theorem is referenced by:  nfiota1  5319  iotaexab  5336  nfrecs  6551  nfsup  7296