Theorem nfco 4712

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 4556	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2282	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2282	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 3982	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2282	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 3982	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1545	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1617	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4004	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2279	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 103   E.wex 1469   F/_wnfc 2269   class class class wbr 3937   {copab 3996   o. ccom 4551

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-co 4556

This theorem is referenced by:  nffun  5154  nftpos  6184  cnmpt11  12491  cnmpt21  12499