Theorem bdph 9970

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 9966	. . . 4 |- BOUNDED y e. { x | ph }
3		df-clab 2027	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	2, 3	bd0 9944	. . 3 |- BOUNDED [ y / x ] ph
5	4	ax-bdsb 9942	. 2 |- BOUNDED [ x / y ] [ y / x ] ph
6		sbid2v 1872	. 2 |- ( [ x / y ] [ y / x ] ph <-> ph )
7	5, 6	bd0 9944	1 |- BOUNDED ph

Syntax hints:   e. wcel 1393   [wsb 1645   {cab 2026  BOUNDED wbd 9932  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-bd0 9933  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-bdc 9961

This theorem is referenced by:  bds  9971