Theorem nfco 4776

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 4620	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2312	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2312	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4035	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2312	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4035	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1558	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1630	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4057	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2309	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 103   E.wex 1485   F/_wnfc 2299   class class class wbr 3989   {copab 4049   o. ccom 4615

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-co 4620

This theorem is referenced by:  nffun  5221  nftpos  6258  cnmpt11  13077  cnmpt21  13085