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Theorem nfovd 7387
Description: Deduction version of bound-variable hypothesis builder nfov 7388. (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 7361 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 nfovd.3 . . 3 (𝜑𝑥𝐹)
3 nfovd.2 . . . 4 (𝜑𝑥𝐴)
4 nfovd.4 . . . 4 (𝜑𝑥𝐵)
53, 4nfopd 4848 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
62, 5nffvd 6855 . 2 (𝜑𝑥(𝐹‘⟨𝐴, 𝐵⟩))
71, 6nfcxfrd 2907 1 (𝜑𝑥(𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnfc 2888  cop 4593  cfv 6497  (class class class)co 7358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  nfov  7388  nfnegd  11397
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