Theorem bdph 15055

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 15051	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2176	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 15029	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 15027	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 2008	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 15029	1 |- BOUNDED ph

Syntax hints:   [wsb 1773   e. wcel 2160   {cab 2175  BOUNDED wbd 15017  BOUNDED wbdc 15045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-bd0 15018  ax-bdsb 15027

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-bdc 15046

This theorem is referenced by:  bds  15056