Theorem nfiunxy 4033

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 4009	. 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 2386	. . . 4 |- F/ y z e. B
5	2, 4	nfrexw 2589	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2397	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2389	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 2209   {cab 2224   F/_wnfc 2379   E.wrex 2529   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-iun 4009

This theorem is referenced by:  iunab  4054