Theorem nfiu1 3916

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 3888	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2520	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2324	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2316	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 2148   {cab 2163   F/_wnfc 2306   E.wrex 2456   U_ciun 3886

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-iun 3888

This theorem is referenced by:  ssiun2s  3930  triun  4113  eliunxp  4764  opeliunxp2  4765  opeliunxp2f  6235  ixpf  6716  ctiunctlemfo  12431