Theorem bdcab 15649

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 15638	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 15647	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2190  BOUNDED wbd 15612  BOUNDED wbdc 15640

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 1471  ax-bd0 15613  ax-bdsb 15622

This theorem depends on definitions:  df-bi 117  df-clab 2191  df-bdc 15641

This theorem is referenced by:  bds  15651  bdcrab  15652  bdccsb  15660  bdcdif  15661  bdcun  15662  bdcin  15663  bdcpw  15669  bdcsn  15670  bdcuni  15676  bdcint  15677  bdciun  15678  bdciin  15679  bdcriota  15683  bj-bdfindis  15747