Theorem bdph 13037

Description: A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdph.1	|- BOUNDED { x | ph }

Assertion

Ref	Expression
bdph	|- BOUNDED ph

Proof of Theorem bdph

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdph.1	. . . . 5 |- BOUNDED { x | ph }
2	1	bdeli 13033	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2124	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 13011	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 13009	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 1969	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 13011	1 |- BOUNDED ph

Syntax hints:   e. wcel 1480   [wsb 1735   {cab 2123  BOUNDED wbd 12999  BOUNDED wbdc 13027

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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-bd0 13000  ax-bdsb 13009

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2124  df-bdc 13028

This theorem is referenced by:  bds  13038