Theorem bj-abv 36889

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 1550	. . . . 5 ⊢ ((𝜑 ∧ 𝜑) → ⊤)
2		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ ⊤) → 𝜑)
3	1, 2	impbida 800	. . . 4 ⊢ (𝜑 → (𝜑 ↔ ⊤))
4	3	alimi 1811	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5		abbi 2795	. . 3 ⊢ (∀𝑥(𝜑 ↔ ⊤) → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
6	4, 5	syl 17	. 2 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
7		dfv2 3453	. 2 ⊢ V = {𝑥 ∣ ⊤}
8	6, 7	eqtr4di 2783	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ⊤wtru 1541  {cab 2708  Vcvv 3450

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 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-v 3452

This theorem is referenced by:  curryset  36929  currysetlem3  36932