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Theorem abv 3477
Description: The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 36276) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2802, ax-8 2100. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
Assertion
Ref Expression
abv ({𝑥𝜑} = V ↔ ∀𝑥𝜑)

Proof of Theorem abv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2717 . . 3 ({𝑥𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2 vextru 2708 . . . . . 6 𝑦 ∈ {𝑥 ∣ ⊤}
32tbt 369 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ (𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
4 df-clab 2702 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
53, 4bitr3i 277 . . . 4 ((𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ [𝑦 / 𝑥]𝜑)
65albii 1813 . . 3 (∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
71, 6bitri 275 . 2 ({𝑥𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
8 dfv2 3469 . . 3 V = {𝑥 ∣ ⊤}
98eqeq2i 2737 . 2 ({𝑥𝜑} = V ↔ {𝑥𝜑} = {𝑥 ∣ ⊤})
10 sb8v 2340 . 2 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
117, 9, 103bitr4i 303 1 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1531   = wceq 1533  wtru 1534  [wsb 2059  wcel 2098  {cab 2701  Vcvv 3466
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-11 2146  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-v 3468
This theorem is referenced by:  dfnf5  4369
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