Theorem bj-abv 37339

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 1564	. . . . 5 ⊢ ((𝜑 ∧ 𝜑) → ⊤)
2		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ ⊤) → 𝜑)
3	1, 2	impbida 808	. . . 4 ⊢ (𝜑 → (𝜑 ↔ ⊤))
4	3	alimi 1825	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤))
5		abbi 2821	. . 3 ⊢ (∀𝑥(𝜑 ↔ ⊤) → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
6	4, 5	syl 17	. 2 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
7		dfv2 3451	. 2 ⊢ V = {𝑥 ∣ ⊤}
8	6, 7	eqtr4di 2809	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1552   = wceq 1554  ⊤wtru 1555  {cab 2734  Vcvv 3448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-v 3450

This theorem is referenced by:  curryset  37379  currysetlem3  37382