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Mirrors > Home > MPE Home > Th. List > abv | Structured version Visualization version GIF version |
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 34120) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) |
Ref | Expression |
---|---|
abv | ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2797 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
2 | 1 | albii 1811 | . 2 ⊢ (∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
3 | eqv 3500 | . 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
4 | nfv 1906 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
5 | 4 | sb8v 2364 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
6 | 2, 3, 5 | 3bitr4i 304 | 1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wal 1526 = wceq 1528 [wsb 2060 ∈ wcel 2105 {cab 2796 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 |
This theorem is referenced by: dfnf5 4331 |
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