Theorem nfco 4699

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 4543	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2279	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2279	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 3969	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2279	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 3969	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1544	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1616	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 3991	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2276	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 103   E.wex 1468   F/_wnfc 2266   class class class wbr 3924   {copab 3983   o. ccom 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-co 4543

This theorem is referenced by:  nffun  5141  nftpos  6169  cnmpt11  12441  cnmpt21  12449