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Theorem abv 3447
Description: The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 33794) requires fewer axioms. (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 2776 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
21albii 1801 . 2 (∀𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3 eqv 3445 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 nfv 1892 . . 3 𝑦𝜑
54sb8v 2329 . 2 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
62, 3, 53bitr4i 304 1 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1520   = wceq 1522  [wsb 2042  wcel 2081  {cab 2775  Vcvv 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-v 3439
This theorem is referenced by:  dfnf5  4254
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