Theorem bdnth 16872

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

Hypothesis

Ref	Expression
bdnth.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
bdnth	⊢ BOUNDED 𝜑

Proof of Theorem bdnth

Step	Hyp	Ref	Expression
1		bdfal 16871	. 2 ⊢ BOUNDED ⊥
2		fal 1409	. . 3 ⊢ ¬ ⊥
3		bdnth.1	. . 3 ⊢ ¬ 𝜑
4	2, 3	2false 713	. 2 ⊢ (⊥ ↔ 𝜑)
5	1, 4	bd0 16862	1 ⊢ BOUNDED 𝜑

This proof depends on syntax axioms:  ¬ wn 3  ⊥wfal 1407  BOUNDED wbd 16850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-bd0 16851  ax-bdim 16852  ax-bdn 16855  ax-bdeq 16858

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is used by:  bdcnul  16903