Theorem nfiunxy 3990

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiunxy.1	|- F/_ y A
nfiunxy.2	|- F/_ y B

Assertion

Ref	Expression
nfiunxy	|- F/_ y U_ x e. A B

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)

Proof of Theorem nfiunxy

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3966	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
2		nfiunxy.1	. . . 4 |- F/_ y A
3		nfiunxy.2	. . . . 5 |- F/_ y B
4	3	nfcri 2366	. . . 4 |- F/ y z e. B
5	2, 4	nfrexw 2569	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2377	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2369	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 2200   {cab 2215   F/_wnfc 2359   E.wrex 2509   U_ciun 3964

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-iun 3966

This theorem is referenced by:  iunab  4011