Theorem bdcab 15718

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 15707	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 15716	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2190  BOUNDED wbd 15681  BOUNDED wbdc 15709

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-gen 1471  ax-bd0 15682  ax-bdsb 15691

This theorem depends on definitions:  df-bi 117  df-clab 2191  df-bdc 15710

This theorem is referenced by:  bds  15720  bdcrab  15721  bdccsb  15729  bdcdif  15730  bdcun  15731  bdcin  15732  bdcpw  15738  bdcsn  15739  bdcuni  15745  bdcint  15746  bdciun  15747  bdciin  15748  bdcriota  15752  bj-bdfindis  15816