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Theorem nfco 5100
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.
StepHypRef Expression
1 df-co 4944 . 2  |-  ( A  o.  B )  =  { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
2 nfcv 2611 . . . . . 6  |-  F/_ x
y
3 nfco.2 . . . . . 6  |-  F/_ x B
4 nfcv 2611 . . . . . 6  |-  F/_ x w
52, 3, 4nfbr 4431 . . . . 5  |-  F/ x  y B w
6 nfco.1 . . . . . 6  |-  F/_ x A
7 nfcv 2611 . . . . . 6  |-  F/_ x
z
84, 6, 7nfbr 4431 . . . . 5  |-  F/ x  w A z
95, 8nfan 1863 . . . 4  |-  F/ x
( y B w  /\  w A z )
109nfex 1883 . . 3  |-  F/ x E. w ( y B w  /\  w A z )
1110nfopab 4452 . 2  |-  F/_ x { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
121, 11nfcxfr 2609 1  |-  F/_ x
( A  o.  B
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   E.wex 1587   F/_wnfc 2597   class class class wbr 4387   {copab 4444    o. ccom 4939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-br 4388  df-opab 4446  df-co 4944
This theorem is referenced by:  nffun  5535  nftpos  6877  cnmpt11  19349  cnmpt21  19357  stoweidlem31  29961  stoweidlem59  29989
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