Theorem nfco 4895

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 4734	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2374	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2374	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4135	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2374	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4135	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1613	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1685	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4157	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2371	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1540   F/_wnfc 2361   class class class wbr 4088   {copab 4149   o. ccom 4729

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-co 4734

This theorem is referenced by:  nffun  5349  nftpos  6444  cnmpt11  15006  cnmpt21  15014