Theorem vexwt 2752

Description: A standard theorem of predicate calculus (stdpc4 2105) expressed using class abstractions. Closed form of vexw 2753. (Contributed by BJ, 14-Jun-2019.)

Assertion

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

Proof of Theorem vexwt

Step	Hyp	Ref	Expression
1		stdpc4 2105	. 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		df-clab 2748	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylibr 237	1 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1565  [wsb 2097   ∈ wcel 2149  {cab 2747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937

This theorem depends on definitions:  df-bi 210  df-an 401  df-sb 2098  df-clab 2748

This theorem is referenced by:  bj-issetwt  37395  bj-abvALT  37427