Theorem bdph 13885

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 13881	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2157	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 13859	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 13857	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 1989	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 13859	1 |- BOUNDED ph

Syntax hints:   [wsb 1755   e. wcel 2141   {cab 2156  BOUNDED wbd 13847  BOUNDED wbdc 13875

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-bd0 13848  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-bdc 13876

This theorem is referenced by:  bds  13886