Theorem bj-abv 37082

Description: The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-abv	⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

Proof of Theorem bj-abv

Step	Hyp	Ref	Expression
1		trud 1552	. . . . 5 ⊢ ((𝜑 ∧ 𝜑) → ⊤)
2		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ ⊤) → 𝜑)
3	1, 2	impbida 801	. . . 4 ⊢ (𝜑 → (𝜑 ↔ ⊤))
4	3	alimi 1813	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5		abbi 2802	. . 3 ⊢ (∀𝑥(𝜑 ↔ ⊤) → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
6	4, 5	syl 17	. 2 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
7		dfv2 3444	. 2 ⊢ V = {𝑥 ∣ ⊤}
8	6, 7	eqtr4di 2790	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ⊤wtru 1543  {cab 2715  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-v 3443

This theorem is referenced by:  curryset  37122  currysetlem3  37125