Theorem bdcab 15603

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 15592	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 15601	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2182  BOUNDED wbd 15566  BOUNDED wbdc 15594

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 15567  ax-bdsb 15576

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-bdc 15595

This theorem is referenced by:  bds  15605  bdcrab  15606  bdccsb  15614  bdcdif  15615  bdcun  15616  bdcin  15617  bdcpw  15623  bdcsn  15624  bdcuni  15630  bdcint  15631  bdciun  15632  bdciin  15633  bdcriota  15637  bj-bdfindis  15701