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Theorem dfafv2 39755
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 39739 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V)
2 df-fv 5697 . . . 4 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
32eqcomi 2523 . . 3 (℩𝑥𝐴𝐹𝑥) = (𝐹𝐴)
4 ifeq1 3943 . . 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 2536 1 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  Vcvv 3077  ifcif 3939   class class class wbr 4481  cio 5651  cfv 5689   defAt wdfat 39735  '''cafv 39736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rab 2809  df-v 3079  df-un 3449  df-if 3940  df-fv 5697  df-afv 39739
This theorem is referenced by:  afveq12d  39756  nfafv  39759  afvfundmfveq  39761  afvnfundmuv  39762  afvpcfv0  39769
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