Theorem bdph 10908

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 10904	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2070	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 10882	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 10880	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 1915	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 10882	1 |- BOUNDED ph

Syntax hints:   e. wcel 1434   [wsb 1687   {cab 2069  BOUNDED wbd 10870  BOUNDED wbdc 10898

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-bd0 10871  ax-bdsb 10880

This theorem depends on definitions:  df-bi 115  df-sb 1688  df-clab 2070  df-bdc 10899

This theorem is referenced by:  bds  10909