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Theorem nfco 4865
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 4714 . 2  |-  ( A  o.  B )  =  { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
2 nfcv 2432 . . . . . 6  |-  F/_ x
y
3 nfco.2 . . . . . 6  |-  F/_ x B
4 nfcv 2432 . . . . . 6  |-  F/_ x w
52, 3, 4nfbr 4083 . . . . 5  |-  F/ x  y B w
6 nfco.1 . . . . . 6  |-  F/_ x A
7 nfcv 2432 . . . . . 6  |-  F/_ x
z
84, 6, 7nfbr 4083 . . . . 5  |-  F/ x  w A z
95, 8nfan 1783 . . . 4  |-  F/ x
( y B w  /\  w A z )
109nfex 1779 . . 3  |-  F/ x E. w ( y B w  /\  w A z )
1110nfopab 4100 . 2  |-  F/_ x { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
121, 11nfcxfr 2429 1  |-  F/_ x
( A  o.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531   F/_wnfc 2419   class class class wbr 4039   {copab 4092    o. ccom 4709
This theorem is referenced by:  nffun  5293  nftpos  6285  cnmpt11  17373  cnmpt21  17381  stoweidlem31  27883  stoweidlem59  27911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-co 4714
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