Theorem bdph 11186

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 11182	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2072	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 11160	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 11158	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 1917	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 11160	1 |- BOUNDED ph

Syntax hints:   e. wcel 1436   [wsb 1689   {cab 2071  BOUNDED wbd 11148  BOUNDED wbdc 11176

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-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-bd0 11149  ax-bdsb 11158

This theorem depends on definitions:  df-bi 115  df-sb 1690  df-clab 2072  df-bdc 11177

This theorem is referenced by:  bds  11187