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Theorem nfafv 43212
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6673. To prove a deduction version of this analogous to nffvd 6675 is not easily possible because a deduction version of nfdfat 43203 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 43208 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 nfafv.1 . . . 4 𝑥𝐹
3 nfafv.2 . . . 4 𝑥𝐴
42, 3nfdfat 43203 . . 3 𝑥 𝐹 defAt 𝐴
52, 3nffv 6673 . . 3 𝑥(𝐹𝐴)
6 nfcv 2974 . . 3 𝑥V
74, 5, 6nfif 4492 . 2 𝑥if(𝐹 defAt 𝐴, (𝐹𝐴), V)
81, 7nfcxfr 2972 1 𝑥(𝐹'''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2958  Vcvv 3492  ifcif 4463  cfv 6348   defAt wdfat 43192  '''cafv 43193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-aiota 43162  df-dfat 43195  df-afv 43196
This theorem is referenced by:  csbafv12g  43213  nfaov  43255
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