Theorem nfco 5287

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 5123	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2764	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2764	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4699	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2764	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4699	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1828	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 2154	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4718	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2762	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 384   E.wex 1704   F/_wnfc 2751   class class class wbr 4653   {copab 4712   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by:  nffun  5911  nftpos  7387  cnmpt11  21466  cnmpt21  21474  poimirlem16  33425  poimirlem19  33428  csbcog  37941  choicefi  39392  cncficcgt0  40101  volioofmpt  40211  volicofmpt  40214  stoweidlem31  40248  stoweidlem59  40276