Theorem vexwt 2746

Description: A standard theorem of predicate calculus (stdpc4 2102) expressed using class abstractions. Closed form of vexw 2747. (Contributed by BJ, 14-Jun-2019.)

Assertion

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

Proof of Theorem vexwt

Step	Hyp	Ref	Expression
1		stdpc4 2102	. 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		df-clab 2742	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylibr 237	1 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  [wsb 2096   ∈ wcel 2143  {cab 2741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742

This theorem is used by:  bj-issetwt  37538  bj-abvALT  37570