Theorem nfiu1 3843

Description: Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)

Assertion

Ref	Expression
nfiu1	|- F/_ x U_ x e. A B

Proof of Theorem nfiu1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3815	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2476	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2286	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2278	1 |- F/_ x 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-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:  ssiun2s  3857  triun  4039  eliunxp  4678  opeliunxp2  4679  opeliunxp2f  6135  ixpf  6614  ctiunctlemfo  11952