Theorem bdab 16876

Description: Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdab.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdab	⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}

Proof of Theorem bdab

Step	Hyp	Ref	Expression
1		bdab.1	. . 3 ⊢ BOUNDED 𝜑
2	1	ax-bdsb 16860	. 2 ⊢ BOUNDED [𝑥 / 𝑦]𝜑
3		df-clab 2225	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
4	2, 3	bd0r 16863	1 ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}

This proof depends on syntax axioms:  [wsb 1815   ∈ wcel 2209  {cab 2224  BOUNDED wbd 16850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 16851  ax-bdsb 16860

This proof depends on definitions:  df-bi 117  df-clab 2225

This theorem is used by:  bdcab  16887  bdsbcALT  16897