Theorem bdab 13873

Description: Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdab.1	|- BOUNDED ph

Assertion

Ref	Expression
bdab	|- BOUNDED x e. { y | ph }

Proof of Theorem bdab

Step	Hyp	Ref	Expression
1		bdab.1	. . 3 |- BOUNDED ph
2	1	ax-bdsb 13857	. 2 |- BOUNDED [ x / y ] ph
3		df-clab 2157	. 2 |- ( x e. { y | ph } <-> [ x / y ] ph )
4	2, 3	bd0r 13860	1 |- BOUNDED x e. { y | ph }

Syntax hints:   [wsb 1755   e. wcel 2141   {cab 2156  BOUNDED wbd 13847

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-bd0 13848  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-clab 2157

This theorem is referenced by:  bdcab  13884  bdsbcALT  13894