Theorem bdnthALT 12438

Description: Alternate proof of bdnth 12437 not using bdfal 12436. Then, bdfal 12436 can be proved from this theorem, using fal 1303. The total number of proof steps would be 17 (for bdnthALT 12438) + 3 = 20, which is more than 8 (for bdfal 12436) + 9 (for bdnth 12437) = 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 12435	. . 3 ⊢ BOUNDED ⊤
2	1	ax-bdn 12420	. 2 ⊢ BOUNDED ¬ ⊤
3		notnot 597	. . . 4 ⊢ (⊤ → ¬ ¬ ⊤)
4	3	mptru 1305	. . 3 ⊢ ¬ ¬ ⊤
5		bdnth.1	. . 3 ⊢ ¬ 𝜑
6	4, 5	2false 655	. 2 ⊢ (¬ ⊤ ↔ 𝜑)
7	2, 6	bd0 12427	1 ⊢ BOUNDED 𝜑

Syntax hints:  ¬ wn 3  ⊤wtru 1297  BOUNDED wbd 12415

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-bd0 12416  ax-bdim 12417  ax-bdn 12420  ax-bdeq 12423

This theorem depends on definitions:  df-bi 116  df-tru 1299

This theorem is referenced by: (None)