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Theorem abv 3411
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 2804 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
21albii 1904 . 2 (∀𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3 eqv 3408 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 nfv 2005 . . 3 𝑦𝜑
54sb8 2585 . 2 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
62, 3, 53bitr4i 294 1 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1635   = wceq 1637  [wsb 2061  wcel 2157  {cab 2803  Vcvv 3402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-v 3404
This theorem is referenced by:  dfnf5  4164
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