Theorem nfiunxy 3899

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 3875	. 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 2306	. . . 4 |- F/ y z e. B
5	2, 4	nfrexxy 2509	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2317	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2309	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 2141   {cab 2156   F/_wnfc 2299   E.wrex 2449   U_ciun 3873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-iun 3875

This theorem is referenced by:  iunab  3919