Theorem bj-abv 37740

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 1580	. . . . 5 ⊢ ((𝜑 ∧ 𝜑) → ⊤)
2		simpl 488	. . . . 5 ⊢ ((𝜑 ∧ ⊤) → 𝜑)
3	1, 2	impbida 813	. . . 4 ⊢ (𝜑 → (𝜑 ↔ ⊤))
4	3	alimi 1844	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5		abbi 2825	. . 3 ⊢ (∀𝑥(𝜑 ↔ ⊤) → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
6	4, 5	syl 18	. 2 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
7		dfv2 3453	. 2 ⊢ V = {𝑥 ∣ ⊤}
8	6, 7	eqtr4di 2813	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ⊤wtru 1571  {cab 2738  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-v 3452

This theorem is used by:  curryset  37781  currysetlem3  37784