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Theorem dfafv2 39861
Description: Alternative definition of (𝐹'''𝐴) using (𝐹𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
Assertion
Ref Expression
dfafv2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)

Proof of Theorem dfafv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-afv 39846 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V)
2 df-fv 5812 . . . 4 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
32eqcomi 2619 . . 3 (℩𝑥𝐴𝐹𝑥) = (𝐹𝐴)
4 ifeq1 4040 . . 3 ((℩𝑥𝐴𝐹𝑥) = (𝐹𝐴) → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V) = if(𝐹 defAt 𝐴, (𝐹𝐴), V))
53, 4ax-mp 5 . 2 if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
61, 5eqtri 2632 1 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  Vcvv 3173  ifcif 4036   class class class wbr 4583  cio 5766  cfv 5804   defAt wdfat 39842  '''cafv 39843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037  df-fv 5812  df-afv 39846
This theorem is referenced by:  afveq12d  39862  nfafv  39865  afvfundmfveq  39867  afvnfundmuv  39868  afvpcfv0  39875
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