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Theorem nfovd 7461
Description: Deduction version of bound-variable hypothesis builder nfov 7462. (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 7435 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 nfovd.3 . . 3 (𝜑𝑥𝐹)
3 nfovd.2 . . . 4 (𝜑𝑥𝐴)
4 nfovd.4 . . . 4 (𝜑𝑥𝐵)
53, 4nfopd 4889 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
62, 5nffvd 6917 . 2 (𝜑𝑥(𝐹‘⟨𝐴, 𝐵⟩))
71, 6nfcxfrd 2903 1 (𝜑𝑥(𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnfc 2889  cop 4631  cfv 6560  (class class class)co 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  nfov  7462  nfnegd  11504
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