Theorem bdnth 13869

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 13868	. 2 ⊢ BOUNDED ⊥
2		fal 1355	. . 3 ⊢ ¬ ⊥
3		bdnth.1	. . 3 ⊢ ¬ 𝜑
4	2, 3	2false 696	. 2 ⊢ (⊥ ↔ 𝜑)
5	1, 4	bd0 13859	1 ⊢ BOUNDED 𝜑

Syntax hints:  ¬ wn 3  ⊥wfal 1353  BOUNDED wbd 13847

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-in1 609  ax-in2 610  ax-bd0 13848  ax-bdim 13849  ax-bdn 13852  ax-bdeq 13855

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by:  bdcnul  13900