Theorem bdab 10629

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 10613	. 2 |- BOUNDED [ x / y ] ph
3		df-clab 2068	. 2 |- ( x e. { y | ph } <-> [ x / y ] ph )
4	2, 3	bd0r 10616	1 |- BOUNDED x e. { y | ph }

Syntax hints:   e. wcel 1433   [wsb 1685   {cab 2067  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-bd0 10604  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068

This theorem is referenced by:  bdcab  10640  bdsbcALT  10650