Theorem nfco 4922

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 4760	. 2 |- ( A o. B ) = { <. y , z >. | E. w ( y B w /\ w A z ) }
2		nfcv 2386	. . . . . 6 |- F/_ x y
3		nfco.2	. . . . . 6 |- F/_ x B
4		nfcv 2386	. . . . . 6 |- F/_ x w
5	2, 3, 4	nfbr 4158	. . . . 5 |- F/ x y B w
6		nfco.1	. . . . . 6 |- F/_ x A
7		nfcv 2386	. . . . . 6 |- F/_ x z
8	4, 6, 7	nfbr 4158	. . . . 5 |- F/ x w A z
9	5, 8	nfan 1614	. . . 4 |- F/ x ( y B w /\ w A z )
10	9	nfex 1686	. . 3 |- F/ x E. w ( y B w /\ w A z )
11	10	nfopab 4180	. 2 |- F/_ x { <. y , z >. | E. w ( y B w /\ w A z ) }
12	1, 11	nfcxfr 2383	1 |- F/_ x ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1541   F/_wnfc 2373   class class class wbr 4111   {copab 4172   o. ccom 4755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-co 4760

This theorem is referenced by:  nffun  5377  nftpos  6512  cnmpt11  15165  cnmpt21  15173