Theorem bdcab 16565

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 16554	. 2 |- BOUNDED y e. { x | ph }
3	2	bdelir 16563	1 |- BOUNDED { x | ph }

Syntax hints:   {cab 2217  BOUNDED wbd 16528  BOUNDED wbdc 16556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1498  ax-bd0 16529  ax-bdsb 16538

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-bdc 16557

This theorem is referenced by:  bds  16567  bdcrab  16568  bdccsb  16576  bdcdif  16577  bdcun  16578  bdcin  16579  bdcpw  16585  bdcsn  16586  bdcuni  16592  bdcint  16593  bdciun  16594  bdciin  16595  bdcriota  16599  bj-bdfindis  16663