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Theorem nfco 5157
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 4997 . 2  |-  ( A  o.  B )  =  { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
2 nfcv 2616 . . . . . 6  |-  F/_ x
y
3 nfco.2 . . . . . 6  |-  F/_ x B
4 nfcv 2616 . . . . . 6  |-  F/_ x w
52, 3, 4nfbr 4483 . . . . 5  |-  F/ x  y B w
6 nfco.1 . . . . . 6  |-  F/_ x A
7 nfcv 2616 . . . . . 6  |-  F/_ x
z
84, 6, 7nfbr 4483 . . . . 5  |-  F/ x  w A z
95, 8nfan 1933 . . . 4  |-  F/ x
( y B w  /\  w A z )
109nfex 1953 . . 3  |-  F/ x E. w ( y B w  /\  w A z )
1110nfopab 4504 . 2  |-  F/_ x { <. y ,  z
>.  |  E. w
( y B w  /\  w A z ) }
121, 11nfcxfr 2614 1  |-  F/_ x
( A  o.  B
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367   E.wex 1617   F/_wnfc 2602   class class class wbr 4439   {copab 4496    o. ccom 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-co 4997
This theorem is referenced by:  nffun  5592  nftpos  6982  cnmpt11  20330  cnmpt21  20338  cncficcgt0  31930  stoweidlem31  32052  stoweidlem59  32080
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