Theorem nfco 4893

Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)

Hypotheses

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

Assertion

Ref	Expression
nfco	|- F/_ x ( A o. B )

Proof of Theorem nfco

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-co 4732	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2372	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2372	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4133	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2372	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4133	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1611	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1683	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4155	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2369	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1538   F/_wnfc 2359   class class class wbr 4086   {copab 4147   o. ccom 4727

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-co 4732

This theorem is referenced by:  nffun  5347  nftpos  6440  cnmpt11  14997  cnmpt21  15005