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Theorem nfovd 5637
Description: Deduction version of bound-variable hypothesis builder nfov 5638. (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 5618 . 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 3624 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
62, 5nffvd 5282 . 2  |-  ( ph  -> 
F/_ x ( F `
 <. A ,  B >. ) )
71, 6nfcxfrd 2223 1  |-  ( ph  -> 
F/_ x ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/_wnfc 2212   <.cop 3434   ` cfv 4983  (class class class)co 5615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-iota 4948  df-fv 4991  df-ov 5618
This theorem is referenced by:  nfov  5638  nfnegd  7625
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