Theorem abv 3504

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 34218) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.)

Assertion

Ref	Expression
abv	⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Proof of Theorem abv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clab 2800	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2	1	albii 1816	. 2 ⊢ (∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3		eqv 3502	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
4		nfv 1911	. . 3 ⊢ Ⅎ𝑦𝜑
5	4	sb8v 2369	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
6	2, 3, 5	3bitr4i 305	1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1531   = wceq 1533  [wsb 2065   ∈ wcel 2110  {cab 2799  Vcvv 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496

This theorem is referenced by:  dfnf5  4333