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Theorem bj-abv 32601
 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 1836 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 bj-vexwvt 32556 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1748 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3195 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 224 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478   = wceq 1480   ∈ wcel 1987  {cab 2607  Vcvv 3190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192 This theorem is referenced by: (None)
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