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Theorem nfovd 5897
Description: Deduction version of bound-variable hypothesis builder nfov 5898. (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 5871 . 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 3793 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
62, 5nffvd 5522 . 2  |-  ( ph  -> 
F/_ x ( F `
 <. A ,  B >. ) )
71, 6nfcxfrd 2317 1  |-  ( ph  -> 
F/_ x ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/_wnfc 2306   <.cop 3594   ` cfv 5211  (class class class)co 5868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5173  df-fv 5219  df-ov 5871
This theorem is referenced by:  nfov  5898  nfnegd  8130
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