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Theorem nfco 4849
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 4698 . 2  |-  ( A  o.  B )  =  { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
2 nfcv 2419 . . . . . 6  |-  F/_ x
y
3 nfco.2 . . . . . 6  |-  F/_ x B
4 nfcv 2419 . . . . . 6  |-  F/_ x w
52, 3, 4nfbr 4067 . . . . 5  |-  F/ x  y B w
6 nfco.1 . . . . . 6  |-  F/_ x A
7 nfcv 2419 . . . . . 6  |-  F/_ x
z
84, 6, 7nfbr 4067 . . . . 5  |-  F/ x  w A z
95, 8nfan 1771 . . . 4  |-  F/ x
( y B w  /\  w A z )
109nfex 1767 . . 3  |-  F/ x E. w ( y B w  /\  w A z )
1110nfopab 4084 . 2  |-  F/_ x { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
121, 11nfcxfr 2416 1  |-  F/_ x
( A  o.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528   F/_wnfc 2406   class class class wbr 4023   {copab 4076    o. ccom 4693
This theorem is referenced by:  nffun  5277  nftpos  6269  cnmpt11  17357  cnmpt21  17365  stoweidlem31  27780  stoweidlem59  27808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-co 4698
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