Theorem nfco 4861

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 4702	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2350	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2350	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4106	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2350	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4106	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1589	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1661	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4128	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2347	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1516   F/_wnfc 2337   class class class wbr 4059   {copab 4120   o. ccom 4697

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-co 4702

This theorem is referenced by:  nffun  5313  nftpos  6388  cnmpt11  14870  cnmpt21  14878