Theorem nfiu1 4005

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 3977	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2576	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2380	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2372	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 2202   {cab 2217   F/_wnfc 2362   E.wrex 2512   U_ciun 3975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-iun 3977

This theorem is referenced by:  ssiun2s  4019  triun  4205  eliunxp  4875  opeliunxp2  4876  opeliunxp2f  6447  ixpf  6932  ctiunctlemfo  13121