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Theorem nfovd 6660
Description: Deduction version of bound-variable hypothesis builder nfov 6661. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
nfovd.2 (𝜑𝑥𝐴)
nfovd.3 (𝜑𝑥𝐹)
nfovd.4 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfovd (𝜑𝑥(𝐴𝐹𝐵))

Proof of Theorem nfovd
StepHypRef Expression
1 df-ov 6638 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 nfovd.3 . . 3 (𝜑𝑥𝐹)
3 nfovd.2 . . . 4 (𝜑𝑥𝐴)
4 nfovd.4 . . . 4 (𝜑𝑥𝐵)
53, 4nfopd 4410 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
62, 5nffvd 6187 . 2 (𝜑𝑥(𝐹‘⟨𝐴, 𝐵⟩))
71, 6nfcxfrd 2761 1 (𝜑𝑥(𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnfc 2749  cop 4174  cfv 5876  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638
This theorem is referenced by:  nfov  6661  nfnegd  10261
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