Theorem bj-abv 37569

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 1579	. . . . 5 ⊢ ((𝜑 ∧ 𝜑) → ⊤)
2		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ ⊤) → 𝜑)
3	1, 2	impbida 812	. . . 4 ⊢ (𝜑 → (𝜑 ↔ ⊤))
4	3	alimi 1840	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5		abbi 2827	. . 3 ⊢ (∀𝑥(𝜑 ↔ ⊤) → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
6	4, 5	syl 18	. 2 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
7		dfv2 3457	. 2 ⊢ V = {𝑥 ∣ ⊤}
8	6, 7	eqtr4di 2815	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567   = wceq 1569  ⊤wtru 1570  {cab 2740  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-v 3456

This theorem is used by:  curryset  37610  currysetlem3  37613