Theorem nfuni 3845

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 3841	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfuni.1	. . . 4 |- F/_ x A
3		nfv 1542	. . . 4 |- F/ x y e. z
4	2, 3	nfrexw 2536	. . 3 |- F/ x E. z e. A y e. z
5	4	nfab 2344	. 2 |- F/_ x { y | E. z e. A y e. z }
6	1, 5	nfcxfr 2336	1 |- F/_ x U. A

Syntax hints:   {cab 2182   F/_wnfc 2326   E.wrex 2476   U.cuni 3839

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-uni 3840

This theorem is referenced by:  nfiota1  5221  iotaexab  5237  nfrecs  6365  nfsup  7056