Theorem nfiu1 3942

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 3914	. 2 |- U_ x e. A B = { y | E. x e. A y e. B }
2		nfre1 2537	. . 3 |- F/ x E. x e. A y e. B
3	2	nfab 2341	. 2 |- F/_ x { y | E. x e. A y e. B }
4	1, 3	nfcxfr 2333	1 |- F/_ x U_ x e. A B

Syntax hints:   e. wcel 2164   {cab 2179   F/_wnfc 2323   E.wrex 2473   U_ciun 3912

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-iun 3914

This theorem is referenced by:  ssiun2s  3956  triun  4140  eliunxp  4801  opeliunxp2  4802  opeliunxp2f  6291  ixpf  6774  ctiunctlemfo  12596