Theorem bdcab 10640

Description: A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdcab.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdcab	⊢ BOUNDED {𝑥 ∣ 𝜑}

Proof of Theorem bdcab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcab.1	. . 3 ⊢ BOUNDED 𝜑
2	1	bdab 10629	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 10638	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2067  BOUNDED wbd 10603  BOUNDED wbdc 10631

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-gen 1378  ax-bd0 10604  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-bdc 10632

This theorem is referenced by:  bds  10642  bdcrab  10643  bdccsb  10651  bdcdif  10652  bdcun  10653  bdcin  10654  bdcpw  10660  bdcsn  10661  bdcuni  10667  bdcint  10668  bdciun  10669  bdciin  10670  bdcriota  10674  bj-bdfindis  10742