Theorem bdeq 17015

Description: Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdeq.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bdeq	⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓)

Proof of Theorem bdeq

Step	Hyp	Ref	Expression
1		bdeq.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	ax-bd0 17005	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
3	1	bicomi 132	. . 3 ⊢ (𝜓 ↔ 𝜑)
4	3	ax-bd0 17005	. 2 ⊢ (BOUNDED 𝜓 → BOUNDED 𝜑)
5	2, 4	impbii 126	1 ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓)

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 17004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 17005

This proof depends on definitions:  df-bi 117

This theorem is used by:  bdceq  17034