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Theorem nfafv 40550
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6165. To prove a deduction version of this analogous to nffvd 6167 is not easily possible because a deduction version of nfdfat 40544 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 40546 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 nfafv.1 . . . 4 𝑥𝐹
3 nfafv.2 . . . 4 𝑥𝐴
42, 3nfdfat 40544 . . 3 𝑥 𝐹 defAt 𝐴
52, 3nffv 6165 . . 3 𝑥(𝐹𝐴)
6 nfcv 2761 . . 3 𝑥V
74, 5, 6nfif 4093 . 2 𝑥if(𝐹 defAt 𝐴, (𝐹𝐴), V)
81, 7nfcxfr 2759 1 𝑥(𝐹'''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2748  Vcvv 3190  ifcif 4064  cfv 5857   defAt wdfat 40527  '''cafv 40528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865  df-dfat 40530  df-afv 40531
This theorem is referenced by:  csbafv12g  40551  nfaov  40593
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