Theorem bdcab 16973

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 16962	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 16971	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:  {cab 2224  BOUNDED wbd 16936  BOUNDED wbdc 16964

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 16937  ax-bdsb 16946

This proof depends on definitions:  df-bi 117  df-clab 2225  df-bdc 16965

This theorem is used by:  bds  16975  bdcrab  16976  bdccsb  16984  bdcdif  16985  bdcun  16986  bdcin  16987  bdcpw  16993  bdcsn  16994  bdcuni  17000  bdcint  17001  bdciun  17002  bdciin  17003  bdcriota  17007  bj-bdfindis  17071