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Theorem bj-abv 34240
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 1911 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 vexwt 2803 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1824 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3481 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 236 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535   = wceq 1537  wcel 2114  {cab 2798  Vcvv 3473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3475
This theorem is referenced by:  curryset  34274  currysetlem3  34277
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