Theorem bdth 13713

Description: A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdth.1	⊢ 𝜑

Assertion

Ref	Expression
bdth	⊢ BOUNDED 𝜑

Proof of Theorem bdth

Step	Hyp	Ref	Expression
1		ax-bdeq 13702	. . 3 ⊢ BOUNDED 𝑥 = 𝑥
2	1, 1	ax-bdim 13696	. 2 ⊢ BOUNDED (𝑥 = 𝑥 → 𝑥 = 𝑥)
3		id 19	. . 3 ⊢ (𝑥 = 𝑥 → 𝑥 = 𝑥)
4		bdth.1	. . 3 ⊢ 𝜑
5	3, 4	2th 173	. 2 ⊢ ((𝑥 = 𝑥 → 𝑥 = 𝑥) ↔ 𝜑)
6	2, 5	bd0 13706	1 ⊢ BOUNDED 𝜑

Syntax hints:   → wi 4  BOUNDED wbd 13694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107  ax-bd0 13695  ax-bdim 13696  ax-bdeq 13702

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bdtru  13714  bdcvv  13739