Theorem nfiunya 3841

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

Hypotheses

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

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem nfiunya

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3815	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
2		nfiunya.1	. . . 4 |- F/_ y A
3		nfiunya.2	. . . . 5 |- F/_ y B
4	3	nfcri 2275	. . . 4 |- F/ y z e. B
5	2, 4	nfrexya 2474	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2286	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2278	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 1480   {cab 2125   F/_wnfc 2268   E.wrex 2417   U_ciun 3813

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-iun 3815

This theorem is referenced by: (None)