Theorem bdph 13732

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 13728	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2152	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 13706	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 13704	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 1984	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 13706	1 |- BOUNDED ph

Syntax hints:   [wsb 1750   e. wcel 2136   {cab 2151  BOUNDED wbd 13694  BOUNDED wbdc 13722

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-bd0 13695  ax-bdsb 13704

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-bdc 13723

This theorem is referenced by:  bds  13733