Theorem bdcab 9969

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 9958	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 9967	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2026  BOUNDED wbd 9932  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-bd0 9933  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-bdc 9961

This theorem is referenced by:  bds  9971  bdcrab  9972  bdccsb  9980  bdcdif  9981  bdcun  9982  bdcin  9983  bdcpw  9989  bdcsn  9990  bdcuni  9996  bdcint  9997  bdciun  9998  bdciin  9999  bdcriota  10003  bj-bdfindis  10072