Theorem bdcab 17080

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 17069	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 17078	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:  {cab 2224  BOUNDED wbd 17043  BOUNDED wbdc 17071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-bd0 17044  ax-bdsb 17053

This proof depends on definitions:  df-bi 117  df-clab 2225  df-bdc 17072

This theorem is used by:  bds  17082  bdcrab  17083  bdccsb  17091  bdcdif  17092  bdcun  17093  bdcin  17094  bdcpw  17100  bdcsn  17101  bdcuni  17107  bdcint  17108  bdciun  17109  bdciin  17110  bdcriota  17114  bj-bdfindis  17178