Theorem bdth 13029

Description: A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdth.1	|- ph

Assertion

Ref	Expression
bdth	|- BOUNDED ph

Proof of Theorem bdth

Step	Hyp	Ref	Expression
1		ax-bdeq 13018	. . 3 |- BOUNDED x = x
2	1, 1	ax-bdim 13012	. 2 |- BOUNDED ( x = x -> x = x )
3		id 19	. . 3 |- ( x = x -> x = x )
4		bdth.1	. . 3 |- ph
5	3, 4	2th 173	. 2 |- ( ( x = x -> x = x ) <-> ph )
6	2, 5	bd0 13022	1 |- BOUNDED ph

Syntax hints:   -> wi 4  BOUNDED wbd 13010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107  ax-bd0 13011  ax-bdim 13012  ax-bdeq 13018

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdtru  13030  bdcvv  13055