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Theorem nfovd 5896
Description: Deduction version of bound-variable hypothesis builder nfov 5897. (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 5877 . 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 3829 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
62, 5nffvd 5550 . 2  |-  ( ph  -> 
F/_ x ( F `
 <. A ,  B >. ) )
71, 6nfcxfrd 2430 1  |-  ( ph  -> 
F/_ x ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/_wnfc 2419   <.cop 3656   ` cfv 5271  (class class class)co 5874
This theorem is referenced by:  nfov  5897  nfnegd  9063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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