Theorem nfco 4794

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 4637	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2319	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2319	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4051	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2319	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4051	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1565	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1637	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4073	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2316	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1492   F/_wnfc 2306   class class class wbr 4005   {copab 4065   o. ccom 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-co 4637

This theorem is referenced by:  nffun  5241  nftpos  6282  cnmpt11  13868  cnmpt21  13876