Theorem bdab 13873

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 13857	. 2 ⊢ BOUNDED [𝑥 / 𝑦]𝜑
3		df-clab 2157	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
4	2, 3	bd0r 13860	1 ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}

Syntax hints:  [wsb 1755   ∈ wcel 2141  {cab 2156  BOUNDED wbd 13847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13848  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-clab 2157

This theorem is referenced by:  bdcab  13884  bdsbcALT  13894