Theorem bdab 10629

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 10613	. 2 ⊢ BOUNDED [𝑥 / 𝑦]𝜑
3		df-clab 2068	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
4	2, 3	bd0r 10616	1 ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}

Syntax hints:   ∈ wcel 1433  [wsb 1685  {cab 2067  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-bd0 10604  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068

This theorem is referenced by:  bdcab  10640  bdsbcALT  10650