Theorem nfiu1 3708

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 3680	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2407	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2223	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2216	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 1433   {cab 2067   F/_wnfc 2206   E.wrex 2349   U_ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-iun 3680

This theorem is referenced by:  ssiun2s  3722  triun  3888  eliunxp  4493  opeliunxp2  4494