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Theorem nfovd 6095
Description: Deduction version of bound-variable hypothesis builder nfov 6096. (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 6076 . 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 3993 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
62, 5nffvd 5729 . 2  |-  ( ph  -> 
F/_ x ( F `
 <. A ,  B >. ) )
71, 6nfcxfrd 2569 1  |-  ( ph  -> 
F/_ x ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/_wnfc 2558   <.cop 3809   ` cfv 5446  (class class class)co 6073
This theorem is referenced by:  nfov  6096  nfnegd  9293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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