Theorem abv 3409

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 34778) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2809, ax-8 2114. (Revised by Gino Giotto, 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 2729	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2		vextru 2721	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
3	2	tbt 373	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
4		df-clab 2715	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5	3, 4	bitr3i 280	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ [𝑦 / 𝑥]𝜑)
6	5	albii 1827	. . 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 3401	. . 3 ⊢ V = {𝑥 ∣ ⊤}
9	8	eqeq2i 2749	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
10		nfv 1922	. . 3 ⊢ Ⅎ𝑦𝜑
11	10	sb8v 2354	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
12	7, 9, 11	3bitr4i 306	1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1541   = wceq 1543  ⊤wtru 1544  [wsb 2072   ∈ wcel 2112  {cab 2714  Vcvv 3398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-v 3400

This theorem is referenced by:  dfnf5  4278