Theorem vexwt 2748

Description: A standard theorem of predicate calculus (stdpc4 2105) expressed using class abstractions. Closed form of vexw 2749. (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 2744	. 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 2099   ∈ wcel 2146  {cab 2743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744

This theorem is used by:  bj-issetwt  37571  bj-abvALT  37603