Theorem bd3an 13712

Description: A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd3or.1	|- BOUNDED ph
bd3or.2	|- BOUNDED ps
bd3or.3	|- BOUNDED ch

Assertion

Ref	Expression
bd3an	|- BOUNDED ( ph /\ ps /\ ch )

Proof of Theorem bd3an

Step	Hyp	Ref	Expression
1		bd3or.1	. . . 4 |- BOUNDED ph
2		bd3or.2	. . . 4 |- BOUNDED ps
3	1, 2	ax-bdan 13697	. . 3 |- BOUNDED ( ph /\ ps )
4		bd3or.3	. . 3 |- BOUNDED ch
5	3, 4	ax-bdan 13697	. 2 |- BOUNDED ( ( ph /\ ps ) /\ ch )
6		df-3an 970	. 2 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
7	5, 6	bd0r 13707	1 |- BOUNDED ( ph /\ ps /\ ch )

Syntax hints:   /\ wa 103   /\ w3a 968  BOUNDED wbd 13694

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 13695  ax-bdan 13697

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by: (None)