Theorem nfiu1 3851

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 3823	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2479	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2287	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2279	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 1481   {cab 2126   F/_wnfc 2269   E.wrex 2418   U_ciun 3821

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-iun 3823

This theorem is referenced by:  ssiun2s  3865  triun  4047  eliunxp  4686  opeliunxp2  4687  opeliunxp2f  6143  ixpf  6622  ctiunctlemfo  11988