Theorem bdxor 13205

Description: The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

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

Assertion

Ref	Expression
bdxor	|- BOUNDED ( ph \/_ ps )

Proof of Theorem bdxor

Step	Hyp	Ref	Expression
1		bdxor.1	. . . 4 |- BOUNDED ph
2		bdxor.2	. . . 4 |- BOUNDED ps
3	1, 2	ax-bdor 13185	. . 3 |- BOUNDED ( ph \/ ps )
4	1, 2	ax-bdan 13184	. . . 4 |- BOUNDED ( ph /\ ps )
5	4	ax-bdn 13186	. . 3 |- BOUNDED -. ( ph /\ ps )
6	3, 5	ax-bdan 13184	. 2 |- BOUNDED ( ( ph \/ ps ) /\ -. ( ph /\ ps ) )
7		df-xor 1355	. 2 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
8	6, 7	bd0r 13194	1 |- BOUNDED ( ph \/_ ps )

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 698   \/_ wxo 1354  BOUNDED wbd 13181

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 13182  ax-bdan 13184  ax-bdor 13185  ax-bdn 13186

This theorem depends on definitions:  df-bi 116  df-xor 1355

This theorem is referenced by: (None)