Theorem nfiu1 3960

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 3932	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2550	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2354	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2346	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 2177   {cab 2192   F/_wnfc 2336   E.wrex 2486   U_ciun 3930

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-iun 3932

This theorem is referenced by:  ssiun2s  3974  triun  4160  eliunxp  4822  opeliunxp2  4823  opeliunxp2f  6334  ixpf  6817  ctiunctlemfo  12860