Theorem nfco 4940

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 4778	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2392	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2392	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4172	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2392	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4172	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1618	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1690	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4194	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2389	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1545   F/_wnfc 2379   class class class wbr 4125   {copab 4186   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-co 4778

This theorem is referenced by:  nffun  5395  nftpos  6540  cnmpt11  15307  cnmpt21  15315