Theorem nfiu1 3896

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 3868	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2509	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2313	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2305	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 2136   {cab 2151   F/_wnfc 2295   E.wrex 2445   U_ciun 3866

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-iun 3868

This theorem is referenced by:  ssiun2s  3910  triun  4093  eliunxp  4743  opeliunxp2  4744  opeliunxp2f  6206  ixpf  6686  ctiunctlemfo  12372