Theorem bdcab 15059

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 15048	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 15057	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2175  BOUNDED wbd 15022  BOUNDED wbdc 15050

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 1460  ax-bd0 15023  ax-bdsb 15032

This theorem depends on definitions:  df-bi 117  df-clab 2176  df-bdc 15051

This theorem is referenced by:  bds  15061  bdcrab  15062  bdccsb  15070  bdcdif  15071  bdcun  15072  bdcin  15073  bdcpw  15079  bdcsn  15080  bdcuni  15086  bdcint  15087  bdciun  15088  bdciin  15089  bdcriota  15093  bj-bdfindis  15157