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Theorem nfafv 41537
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6236. To prove a deduction version of this analogous to nffvd 6238 is not easily possible because a deduction version of nfdfat 41531 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 41533 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 nfafv.1 . . . 4 𝑥𝐹
3 nfafv.2 . . . 4 𝑥𝐴
42, 3nfdfat 41531 . . 3 𝑥 𝐹 defAt 𝐴
52, 3nffv 6236 . . 3 𝑥(𝐹𝐴)
6 nfcv 2793 . . 3 𝑥V
74, 5, 6nfif 4148 . 2 𝑥if(𝐹 defAt 𝐴, (𝐹𝐴), V)
81, 7nfcxfr 2791 1 𝑥(𝐹'''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2780  Vcvv 3231  ifcif 4119  cfv 5926   defAt wdfat 41514  '''cafv 41515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-dfat 41517  df-afv 41518
This theorem is referenced by:  csbafv12g  41538  nfaov  41580
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