Theorem nfco 4810

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 4653	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2332	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2332	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4064	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2332	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4064	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1576	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1648	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4086	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2329	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1503   F/_wnfc 2319   class class class wbr 4018   {copab 4078   o. ccom 4648

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 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-co 4653

This theorem is referenced by:  nffun  5258  nftpos  6303  cnmpt11  14235  cnmpt21  14243