Theorem bj-abv 33423

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

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 2011	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		bj-vexwvt 33376	. . 3 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	alrimih 1924	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
4		eqv 3421	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
5	3, 4	sylibr 226	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

Syntax hints:   → wi 4  ∀wal 1656   = wceq 1658   ∈ wcel 2166  {cab 2812  Vcvv 3415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-v 3417

This theorem is referenced by: (None)