Theorem abv 3467

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 37561) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2838, ax-8 2145. (Revised by GG, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)

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		dfcleq 2756	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2		vextru 2748	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
3	2	tbt 372	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
4		df-clab 2742	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5	3, 4	bitr3i 280	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ [𝑦 / 𝑥]𝜑)
6	5	albii 1849	. . 3 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
7	1, 6	bitri 278	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
8		dfv2 3458	. . 3 ⊢ V = {𝑥 ∣ ⊤}
9	8	eqeq2i 2776	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
10		sb8v 2385	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
11	7, 9, 10	3bitr4i 306	1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1568   = wceq 1570  ⊤wtru 1571  [wsb 2096   ∈ wcel 2143  {cab 2741  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-11 2192  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-v 3457

This theorem is referenced by:  dfnf5  4338  mh-setind  37067  ecqmap  39118