Theorem bdab 11375

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 11359	. 2 |- BOUNDED [ x / y ] ph
3		df-clab 2075	. 2 |- ( x e. { y | ph } <-> [ x / y ] ph )
4	2, 3	bd0r 11362	1 |- BOUNDED x e. { y | ph }

Syntax hints:   e. wcel 1438   [wsb 1692   {cab 2074  BOUNDED wbd 11349

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 11350  ax-bdsb 11359

This theorem depends on definitions:  df-bi 115  df-clab 2075

This theorem is referenced by:  bdcab  11386  bdsbcALT  11396