Theorem bdeq 9943

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 9933	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
3	1	bicomi 123	. . 3 ⊢ (𝜓 ↔ 𝜑)
4	3	ax-bd0 9933	. 2 ⊢ (BOUNDED 𝜓 → BOUNDED 𝜑)
5	2, 4	impbii 117	1 ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓)

Syntax hints:   ↔ wb 98  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-bd0 9933

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bdceq  9962