Theorem abv 3471

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 36924) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2809, ax-8 2110. (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 2728	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2		vextru 2720	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
3	2	tbt 369	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
4		df-clab 2714	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5	3, 4	bitr3i 277	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ [𝑦 / 𝑥]𝜑)
6	5	albii 1819	. . 3 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
7	1, 6	bitri 275	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
8		dfv2 3462	. . 3 ⊢ V = {𝑥 ∣ ⊤}
9	8	eqeq2i 2748	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
10		sb8v 2354	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
11	7, 9, 10	3bitr4i 303	1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540  ⊤wtru 1541  [wsb 2064   ∈ wcel 2108  {cab 2713  Vcvv 3459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-11 2157  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-v 3461

This theorem is referenced by:  dfnf5  4357