Theorem nfco 4843

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 4684	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2348	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2348	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4090	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2348	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4090	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1588	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1660	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4112	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2345	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1515   F/_wnfc 2335   class class class wbr 4044   {copab 4104   o. ccom 4679

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-co 4684

This theorem is referenced by:  nffun  5294  nftpos  6365  cnmpt11  14755  cnmpt21  14763