Theorem bdab 16962

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 16946	. 2 |- BOUNDED [ x / y ] ph
3		df-clab 2225	. 2 |- ( x e. { y | ph } <-> [ x / y ] ph )
4	2, 3	bd0r 16949	1 |- BOUNDED x e. { y | ph }

This proof depends on syntax axioms:   [wsb 1815   e. wcel 2209   {cab 2224  BOUNDED wbd 16936

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 16937  ax-bdsb 16946

This proof depends on definitions:  df-bi 117  df-clab 2225

This theorem is used by:  bdcab  16973  bdsbcALT  16983