Theorem bdcab 14571

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 14560	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 14569	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2163  BOUNDED wbd 14534  BOUNDED wbdc 14562

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 1449  ax-bd0 14535  ax-bdsb 14544

This theorem depends on definitions:  df-bi 117  df-clab 2164  df-bdc 14563

This theorem is referenced by:  bds  14573  bdcrab  14574  bdccsb  14582  bdcdif  14583  bdcun  14584  bdcin  14585  bdcpw  14591  bdcsn  14592  bdcuni  14598  bdcint  14599  bdciun  14600  bdciin  14601  bdcriota  14605  bj-bdfindis  14669