Theorem nfun 3198

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 3041	. 2 |- ( A u. B ) = { y | ( y e. A \/ y e. B ) }
2		nfun.1	. . . . 5 |- F/_ x A
3	2	nfcri 2249	. . . 4 |- F/ x y e. A
4		nfun.2	. . . . 5 |- F/_ x B
5	4	nfcri 2249	. . . 4 |- F/ x y e. B
6	3, 5	nfor 1536	. . 3 |- F/ x ( y e. A \/ y e. B )
7	6	nfab 2260	. 2 |- F/_ x { y | ( y e. A \/ y e. B ) }
8	1, 7	nfcxfr 2252	1 |- F/_ x ( A u. B )

Syntax hints:   \/ wo 680   e. wcel 1463   {cab 2101   F/_wnfc 2242   u. cun 3035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-un 3041

This theorem is referenced by:  nfsuc  4290  nfdju  6879