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Theorem bj-abv 33423
 Description: The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-abv (∀𝑥𝜑 → {𝑥𝜑} = V)

Proof of Theorem bj-abv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-5 2011 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 bj-vexwvt 33376 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1924 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3421 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 226 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1656   = wceq 1658   ∈ wcel 2166  {cab 2812  Vcvv 3415 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-ext 2804 This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-v 3417 This theorem is referenced by: (None)
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