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Theorem nfco 5030
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 4879 . 2  |-  ( A  o.  B )  =  { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
2 nfcv 2571 . . . . . 6  |-  F/_ x
y
3 nfco.2 . . . . . 6  |-  F/_ x B
4 nfcv 2571 . . . . . 6  |-  F/_ x w
52, 3, 4nfbr 4248 . . . . 5  |-  F/ x  y B w
6 nfco.1 . . . . . 6  |-  F/_ x A
7 nfcv 2571 . . . . . 6  |-  F/_ x
z
84, 6, 7nfbr 4248 . . . . 5  |-  F/ x  w A z
95, 8nfan 1846 . . . 4  |-  F/ x
( y B w  /\  w A z )
109nfex 1865 . . 3  |-  F/ x E. w ( y B w  /\  w A z )
1110nfopab 4265 . 2  |-  F/_ x { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
121, 11nfcxfr 2568 1  |-  F/_ x
( A  o.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550   F/_wnfc 2558   class class class wbr 4204   {copab 4257    o. ccom 4874
This theorem is referenced by:  nffun  5468  nftpos  6506  cnmpt11  17687  cnmpt21  17695  stoweidlem31  27747  stoweidlem59  27775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-co 4879
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