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Theorem bj-abv 36252
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 1550 . . . . 5 ((𝜑𝜑) → ⊤)
2 simpl 482 . . . . 5 ((𝜑 ∧ ⊤) → 𝜑)
31, 2impbida 798 . . . 4 (𝜑 → (𝜑 ↔ ⊤))
43alimi 1812 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5 abbi 2799 . . 3 (∀𝑥(𝜑 ↔ ⊤) → {𝑥𝜑} = {𝑥 ∣ ⊤})
64, 5syl 17 . 2 (∀𝑥𝜑 → {𝑥𝜑} = {𝑥 ∣ ⊤})
7 dfv2 3476 . 2 V = {𝑥 ∣ ⊤}
86, 7eqtr4di 2789 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1538   = wceq 1540  wtru 1541  {cab 2708  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 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-v 3475
This theorem is referenced by:  curryset  36293  currysetlem3  36296
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