Theorem bdxor 13718

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

Hypotheses

Ref	Expression
bdxor.1	⊢ BOUNDED 𝜑
bdxor.2	⊢ BOUNDED 𝜓

Assertion

Ref	Expression
bdxor	⊢ BOUNDED (𝜑 ⊻ 𝜓)

Proof of Theorem bdxor

Step	Hyp	Ref	Expression
1		bdxor.1	. . . 4 ⊢ BOUNDED 𝜑
2		bdxor.2	. . . 4 ⊢ BOUNDED 𝜓
3	1, 2	ax-bdor 13698	. . 3 ⊢ BOUNDED (𝜑 ∨ 𝜓)
4	1, 2	ax-bdan 13697	. . . 4 ⊢ BOUNDED (𝜑 ∧ 𝜓)
5	4	ax-bdn 13699	. . 3 ⊢ BOUNDED ¬ (𝜑 ∧ 𝜓)
6	3, 5	ax-bdan 13697	. 2 ⊢ BOUNDED ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))
7		df-xor 1366	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
8	6, 7	bd0r 13707	1 ⊢ BOUNDED (𝜑 ⊻ 𝜓)

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698   ⊻ wxo 1365  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  ax-bdor 13698  ax-bdn 13699

This theorem depends on definitions:  df-bi 116  df-xor 1366

This theorem is referenced by: (None)