Theorem bdbi 13013

Description: A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdbi.1	|- BOUNDED ph
bdbi.2	|- BOUNDED ps

Assertion

Ref	Expression
bdbi	|- BOUNDED ( ph <-> ps )

Proof of Theorem bdbi

Step	Hyp	Ref	Expression
1		bdbi.1	. . . 4 |- BOUNDED ph
2		bdbi.2	. . . 4 |- BOUNDED ps
3	1, 2	ax-bdim 13001	. . 3 |- BOUNDED ( ph -> ps )
4	2, 1	ax-bdim 13001	. . 3 |- BOUNDED ( ps -> ph )
5	3, 4	ax-bdan 13002	. 2 |- BOUNDED ( ( ph -> ps ) /\ ( ps -> ph ) )
6		dfbi2 385	. 2 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
7	5, 6	bd0r 13012	1 |- BOUNDED ( ph <-> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  BOUNDED wbd 12999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13000  ax-bdim 13001  ax-bdan 13002

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)