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Theorem nfafv 47148
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6916. To prove a deduction version of this analogous to nffvd 6918 is not easily possible because a deduction version of nfdfat 47139 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
nfafv.1 𝑥𝐹
nfafv.2 𝑥𝐴
Assertion
Ref Expression
nfafv 𝑥(𝐹'''𝐴)

Proof of Theorem nfafv
StepHypRef Expression
1 dfafv2 47144 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 nfafv.1 . . . 4 𝑥𝐹
3 nfafv.2 . . . 4 𝑥𝐴
42, 3nfdfat 47139 . . 3 𝑥 𝐹 defAt 𝐴
52, 3nffv 6916 . . 3 𝑥(𝐹𝐴)
6 nfcv 2905 . . 3 𝑥V
74, 5, 6nfif 4556 . 2 𝑥if(𝐹 defAt 𝐴, (𝐹𝐴), V)
81, 7nfcxfr 2903 1 𝑥(𝐹'''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2890  Vcvv 3480  ifcif 4525  cfv 6561   defAt wdfat 47128  '''cafv 47129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-aiota 47097  df-dfat 47131  df-afv 47132
This theorem is referenced by:  csbafv12g  47149  nfaov  47191
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