Theorem bdnthALT 16456

Description: Alternate proof of bdnth 16455 not using bdfal 16454. Then, bdfal 16454 can be proved from this theorem, using fal 1404. The total number of proof steps would be 17 (for bdnthALT 16456) + 3 = 20, which is more than 8 (for bdfal 16454) + 9 (for bdnth 16455) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bdnth.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
bdnthALT	⊢ BOUNDED 𝜑

Proof of Theorem bdnthALT

Step	Hyp	Ref	Expression
1		bdtru 16453	. . 3 ⊢ BOUNDED ⊤
2	1	ax-bdn 16438	. 2 ⊢ BOUNDED ¬ ⊤
3		notnot 634	. . . 4 ⊢ (⊤ → ¬ ¬ ⊤)
4	3	mptru 1406	. . 3 ⊢ ¬ ¬ ⊤
5		bdnth.1	. . 3 ⊢ ¬ 𝜑
6	4, 5	2false 708	. 2 ⊢ (¬ ⊤ ↔ 𝜑)
7	2, 6	bd0 16445	1 ⊢ BOUNDED 𝜑

Syntax hints:  ¬ wn 3  ⊤wtru 1398  BOUNDED wbd 16433

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 619  ax-in2 620  ax-bd0 16434  ax-bdim 16435  ax-bdn 16438  ax-bdeq 16441

This theorem depends on definitions:  df-bi 117  df-tru 1400

This theorem is referenced by: (None)