Theorem nfun 3289

Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfun.1	|- F/_ x A
nfun.2	|- F/_ x B

Assertion

Ref	Expression
nfun	|- F/_ x ( A u. B )

Proof of Theorem nfun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-un 3131	. 2 |- ( A u. B ) = { y | ( y e. A \/ y e. B ) }
2		nfun.1	. . . . 5 |- F/_ x A
3	2	nfcri 2311	. . . 4 |- F/ x y e. A
4		nfun.2	. . . . 5 |- F/_ x B
5	4	nfcri 2311	. . . 4 |- F/ x y e. B
6	3, 5	nfor 1572	. . 3 |- F/ x ( y e. A \/ y e. B )
7	6	nfab 2322	. 2 |- F/_ x { y | ( y e. A \/ y e. B ) }
8	1, 7	nfcxfr 2314	1 |- F/_ x ( A u. B )

Syntax hints:   \/ wo 708   e. wcel 2146   {cab 2161   F/_wnfc 2304   u. cun 3125

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-un 3131

This theorem is referenced by:  nfsuc  4402  nfdju  7031