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Theorem bj-abv 34859
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 1553 . . . . 5 ((𝜑𝜑) → ⊤)
2 simpl 486 . . . . 5 ((𝜑 ∧ ⊤) → 𝜑)
31, 2impbida 801 . . . 4 (𝜑 → (𝜑 ↔ ⊤))
43alimi 1819 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5 abbi1 2808 . . 3 (∀𝑥(𝜑 ↔ ⊤) → {𝑥𝜑} = {𝑥 ∣ ⊤})
64, 5syl 17 . 2 (∀𝑥𝜑 → {𝑥𝜑} = {𝑥 ∣ ⊤})
7 dfv2 3425 . 2 V = {𝑥 ∣ ⊤}
86, 7eqtr4di 2798 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wtru 1544  {cab 2716  Vcvv 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-v 3424
This theorem is referenced by:  curryset  34903  currysetlem3  34906
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