Theorem bdcab 13047

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 ph

Assertion

Ref	Expression
bdcab	|- BOUNDED { x | ph }

Proof of Theorem bdcab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcab.1	. . 3 |- BOUNDED ph
2	1	bdab 13036	. 2 |- BOUNDED y e. { x | ph }
3	2	bdelir 13045	1 |- BOUNDED { x | ph }

Syntax hints:   {cab 2125  BOUNDED wbd 13010  BOUNDED wbdc 13038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425  ax-bd0 13011  ax-bdsb 13020

This theorem depends on definitions:  df-bi 116  df-clab 2126  df-bdc 13039

This theorem is referenced by:  bds  13049  bdcrab  13050  bdccsb  13058  bdcdif  13059  bdcun  13060  bdcin  13061  bdcpw  13067  bdcsn  13068  bdcuni  13074  bdcint  13075  bdciun  13076  bdciin  13077  bdcriota  13081  bj-bdfindis  13145