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Theorem dfafv2 41718
 Description: Alternative definition of (𝐹'''𝐴) using (𝐹‘𝐴) 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 41703 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V)
2 df-fv 6057 . . . 4 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
32eqcomi 2769 . . 3 (℩𝑥𝐴𝐹𝑥) = (𝐹𝐴)
4 ifeq1 4234 . . 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 2782 1 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632  Vcvv 3340  ifcif 4230   class class class wbr 4804  ℩cio 6010  ‘cfv 6049   defAt wdfat 41699  '''cafv 41700 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-un 3720  df-if 4231  df-fv 6057  df-afv 41703 This theorem is referenced by:  afveq12d  41719  nfafv  41722  afvfundmfveq  41724  afvnfundmuv  41725  afvpcfv0  41732
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