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Theorem nfafv 44082
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6668. To prove a deduction version of this analogous to nffvd 6670 is not easily possible because a deduction version of nfdfat 44073 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 44078 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 nfafv.1 . . . 4 𝑥𝐹
3 nfafv.2 . . . 4 𝑥𝐴
42, 3nfdfat 44073 . . 3 𝑥 𝐹 defAt 𝐴
52, 3nffv 6668 . . 3 𝑥(𝐹𝐴)
6 nfcv 2919 . . 3 𝑥V
74, 5, 6nfif 4450 . 2 𝑥if(𝐹 defAt 𝐴, (𝐹𝐴), V)
81, 7nfcxfr 2917 1 𝑥(𝐹'''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2899  Vcvv 3409  ifcif 4420  cfv 6335   defAt wdfat 44062  '''cafv 44063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-res 5536  df-iota 6294  df-fun 6337  df-fv 6343  df-aiota 44030  df-dfat 44065  df-afv 44066
This theorem is referenced by:  csbafv12g  44083  nfaov  44125
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