Theorem nfuni 3708

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 3704	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfuni.1	. . . 4 |- F/_ x A
3		nfv 1491	. . . 4 |- F/ x y e. z
4	2, 3	nfrexxy 2446	. . 3 |- F/ x E. z e. A y e. z
5	4	nfab 2260	. 2 |- F/_ x { y | E. z e. A y e. z }
6	1, 5	nfcxfr 2252	1 |- F/_ x U. A

Syntax hints:   {cab 2101   F/_wnfc 2242   E.wrex 2391   U.cuni 3702

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-uni 3703

This theorem is referenced by:  nfiota1  5048  nfrecs  6158  nfsup  6831