Theorem nfiu1 3716

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 3688	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2408	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2224	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2217	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 1434   {cab 2068   F/_wnfc 2207   E.wrex 2350   U_ciun 3686

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-iun 3688

This theorem is referenced by:  ssiun2s  3730  triun  3896  eliunxp  4503  opeliunxp2  4504