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Theorem abv 3502
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 34120) 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 2797 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
21albii 1811 . 2 (∀𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3 eqv 3500 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 nfv 1906 . . 3 𝑦𝜑
54sb8v 2364 . 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 1526   = wceq 1528  [wsb 2060  wcel 2105  {cab 2796  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494
This theorem is referenced by:  dfnf5  4331
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