Theorem bdth 17023

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 17012	. . 3 ⊢ BOUNDED 𝑥 = 𝑥
2	1, 1	ax-bdim 17006	. 2 ⊢ BOUNDED (𝑥 = 𝑥 → 𝑥 = 𝑥)
3		id 19	. . 3 ⊢ (𝑥 = 𝑥 → 𝑥 = 𝑥)
4		bdth.1	. . 3 ⊢ 𝜑
5	3, 4	2th 174	. 2 ⊢ ((𝑥 = 𝑥 → 𝑥 = 𝑥) ↔ 𝜑)
6	2, 5	bd0 17016	1 ⊢ BOUNDED 𝜑

This proof depends on syntax axioms:   → wi 4  BOUNDED wbd 17004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-bd0 17005  ax-bdim 17006  ax-bdeq 17012

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdtru  17024  bdcvv  17049