Theorem bdab 15043

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 15027	. 2 ⊢ BOUNDED [𝑥 / 𝑦]𝜑
3		df-clab 2176	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
4	2, 3	bd0r 15030	1 ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}

Syntax hints:  [wsb 1773   ∈ wcel 2160  {cab 2175  BOUNDED wbd 15017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 15018  ax-bdsb 15027

This theorem depends on definitions:  df-bi 117  df-clab 2176

This theorem is referenced by:  bdcab  15054  bdsbcALT  15064