Theorem bdnth 14969

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 14968	. 2 ⊢ BOUNDED ⊥
2		fal 1370	. . 3 ⊢ ¬ ⊥
3		bdnth.1	. . 3 ⊢ ¬ 𝜑
4	2, 3	2false 702	. 2 ⊢ (⊥ ↔ 𝜑)
5	1, 4	bd0 14959	1 ⊢ BOUNDED 𝜑

Syntax hints:  ¬ wn 3  ⊥wfal 1368  BOUNDED wbd 14947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-bd0 14948  ax-bdim 14949  ax-bdn 14952  ax-bdeq 14955

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369

This theorem is referenced by:  bdcnul  15000