Theorem bdcab 17097

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 17086	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 17095	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:  {cab 2224  BOUNDED wbd 17060  BOUNDED wbdc 17088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-bd0 17061  ax-bdsb 17070

This proof depends on definitions:  df-bi 117  df-clab 2225  df-bdc 17089

This theorem is used by:  bds  17099  bdcrab  17100  bdccsb  17108  bdcdif  17109  bdcun  17110  bdcin  17111  bdcpw  17117  bdcsn  17118  bdcuni  17124  bdcint  17125  bdciun  17126  bdciin  17127  bdcriota  17131  bj-bdfindis  17195