Theorem bdnth 9954

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 9953	. 2 ⊢ BOUNDED ⊥
2		fal 1250	. . 3 ⊢ ¬ ⊥
3		bdnth.1	. . 3 ⊢ ¬ 𝜑
4	2, 3	2false 617	. 2 ⊢ (⊥ ↔ 𝜑)
5	1, 4	bd0 9944	1 ⊢ BOUNDED 𝜑

Syntax hints:  ¬ wn 3  ⊥wfal 1248  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdeq 9940

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  bdcnul  9985