Theorem nfco 4769

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 4613	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2308	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2308	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4028	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2308	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4028	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1553	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1625	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4050	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2305	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 103   E.wex 1480   F/_wnfc 2295   class class class wbr 3982   {copab 4042   o. ccom 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-co 4613

This theorem is referenced by:  nffun  5211  nftpos  6247  cnmpt11  12923  cnmpt21  12931