Theorem bdcab 15984

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 15973	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 15982	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2193  BOUNDED wbd 15947  BOUNDED wbdc 15975

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 1473  ax-bd0 15948  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-clab 2194  df-bdc 15976

This theorem is referenced by:  bds  15986  bdcrab  15987  bdccsb  15995  bdcdif  15996  bdcun  15997  bdcin  15998  bdcpw  16004  bdcsn  16005  bdcuni  16011  bdcint  16012  bdciun  16013  bdciin  16014  bdcriota  16018  bj-bdfindis  16082