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Theorem nfco 4810
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 4653 . 2  |-  ( A  o.  B )  =  { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
2 nfcv 2332 . . . . . 6  |-  F/_ x
y
3 nfco.2 . . . . . 6  |-  F/_ x B
4 nfcv 2332 . . . . . 6  |-  F/_ x w
52, 3, 4nfbr 4064 . . . . 5  |-  F/ x  y B w
6 nfco.1 . . . . . 6  |-  F/_ x A
7 nfcv 2332 . . . . . 6  |-  F/_ x
z
84, 6, 7nfbr 4064 . . . . 5  |-  F/ x  w A z
95, 8nfan 1576 . . . 4  |-  F/ x
( y B w  /\  w A z )
109nfex 1648 . . 3  |-  F/ x E. w ( y B w  /\  w A z )
1110nfopab 4086 . 2  |-  F/_ x { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
121, 11nfcxfr 2329 1  |-  F/_ x
( A  o.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104   E.wex 1503   F/_wnfc 2319   class class class wbr 4018   {copab 4078    o. ccom 4648
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-co 4653
This theorem is referenced by:  nffun  5258  nftpos  6303  cnmpt11  14235  cnmpt21  14243
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