Theorem bdcab 15785

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 15774	. 2 |- BOUNDED y e. { x | ph }
3	2	bdelir 15783	1 |- BOUNDED { x | ph }

Syntax hints:   {cab 2191  BOUNDED wbd 15748  BOUNDED wbdc 15776

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 1472  ax-bd0 15749  ax-bdsb 15758

This theorem depends on definitions:  df-bi 117  df-clab 2192  df-bdc 15777

This theorem is referenced by:  bds  15787  bdcrab  15788  bdccsb  15796  bdcdif  15797  bdcun  15798  bdcin  15799  bdcpw  15805  bdcsn  15806  bdcuni  15812  bdcint  15813  bdciun  15814  bdciin  15815  bdcriota  15819  bj-bdfindis  15883