Theorem bdcab 15581

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 15570	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 15579	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2182  BOUNDED wbd 15544  BOUNDED wbdc 15572

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 1463  ax-bd0 15545  ax-bdsb 15554

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-bdc 15573

This theorem is referenced by:  bds  15583  bdcrab  15584  bdccsb  15592  bdcdif  15593  bdcun  15594  bdcin  15595  bdcpw  15601  bdcsn  15602  bdcuni  15608  bdcint  15609  bdciun  15610  bdciin  15611  bdcriota  15615  bj-bdfindis  15679