Theorem vexwt 2781

Description: A standard theorem of predicate calculus (stdpc4 2073) expressed using class abstractions. Closed form of vexw 2782. (Contributed by BJ, 14-Jun-2019.)

Assertion

Ref	Expression
vexwt	⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})

Proof of Theorem vexwt

Step	Hyp	Ref	Expression
1		stdpc4 2073	. 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		df-clab 2777	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylibr 237	1 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1536  [wsb 2069   ∈ wcel 2111  {cab 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911

This theorem depends on definitions:  df-bi 210  df-sb 2070  df-clab 2777

This theorem is referenced by:  bj-issetwt  34313  bj-abv  34347