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Theorem nfovd 5800
Description: Deduction version of bound-variable hypothesis builder nfov 5801. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
nfovd.2  |-  ( ph  -> 
F/_ x A )
nfovd.3  |-  ( ph  -> 
F/_ x F )
nfovd.4  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfovd  |-  ( ph  -> 
F/_ x ( A F B ) )

Proof of Theorem nfovd
StepHypRef Expression
1 df-ov 5777 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 nfovd.3 . . 3  |-  ( ph  -> 
F/_ x F )
3 nfovd.2 . . . 4  |-  ( ph  -> 
F/_ x A )
4 nfovd.4 . . . 4  |-  ( ph  -> 
F/_ x B )
53, 4nfopd 3722 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
62, 5nffvd 5433 . 2  |-  ( ph  -> 
F/_ x ( F `
 <. A ,  B >. ) )
71, 6nfcxfrd 2279 1  |-  ( ph  -> 
F/_ x ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/_wnfc 2268   <.cop 3530   ` cfv 5123  (class class class)co 5774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  nfov  5801  nfnegd  7970
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