Theorem bdph 13219

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 13215	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2127	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 13193	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 13191	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 1972	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 13193	1 |- BOUNDED ph

Syntax hints:   e. wcel 1481   [wsb 1736   {cab 2126  BOUNDED wbd 13181  BOUNDED wbdc 13209

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-bd0 13182  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-bdc 13210

This theorem is referenced by:  bds  13220