Theorem bdph 15985

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 15981	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2194	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 15959	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 15957	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 2025	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 15959	1 |- BOUNDED ph

Syntax hints:   [wsb 1786   e. wcel 2178   {cab 2193  BOUNDED wbd 15947  BOUNDED wbdc 15975

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-bd0 15948  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2194  df-bdc 15976

This theorem is referenced by:  bds  15986