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Theorem bj-abv 36293
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
StepHypRef Expression
1 trud 1543 . . . . 5 ((𝜑𝜑) → ⊤)
2 simpl 482 . . . . 5 ((𝜑 ∧ ⊤) → 𝜑)
31, 2impbida 798 . . . 4 (𝜑 → (𝜑 ↔ ⊤))
43alimi 1805 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5 abbi 2794 . . 3 (∀𝑥(𝜑 ↔ ⊤) → {𝑥𝜑} = {𝑥 ∣ ⊤})
64, 5syl 17 . 2 (∀𝑥𝜑 → {𝑥𝜑} = {𝑥 ∣ ⊤})
7 dfv2 3471 . 2 V = {𝑥 ∣ ⊤}
86, 7eqtr4di 2784 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wtru 1534  {cab 2703  Vcvv 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-v 3470
This theorem is referenced by:  curryset  36334  currysetlem3  36337
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