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Theorem abv 3179
 Description: The class of sets verifying a property is the universal class if and only if that property is a tautology. (Contributed by BJ, 19-Mar-2021.)
Assertion
Ref Expression
abv ({𝑥𝜑} = V ↔ ∀𝑥𝜑)

Proof of Theorem abv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-clab 2597 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
21albii 1737 . 2 (∀𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3 eqv 3178 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 nfv 1830 . . 3 𝑦𝜑
54sb8 2412 . 2 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
62, 3, 53bitr4i 291 1 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596  Vcvv 3173 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-v 3175 This theorem is referenced by:  dfnf5  3906
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