Theorem bdcab 16875

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 16864	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 16873	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:  {cab 2224  BOUNDED wbd 16838  BOUNDED wbdc 16866

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 16839  ax-bdsb 16848

This proof depends on definitions:  df-bi 117  df-clab 2225  df-bdc 16867

This theorem is used by:  bds  16877  bdcrab  16878  bdccsb  16886  bdcdif  16887  bdcun  16888  bdcin  16889  bdcpw  16895  bdcsn  16896  bdcuni  16902  bdcint  16903  bdciun  16904  bdciin  16905  bdcriota  16909  bj-bdfindis  16973