Theorem nfiunxy 3683

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 3659	. 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 2172	. . . 4 |- F/ y z e. B
5	2, 4	nfrexxy 2361	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2182	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2175	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 1393   {cab 2026   F/_wnfc 2165   E.wrex 2307   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-iun 3659

This theorem is referenced by:  iunab  3703