Theorem bdbi 17018

Description: A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdbi.1	⊢ BOUNDED 𝜑
bdbi.2	⊢ BOUNDED 𝜓

Assertion

Ref	Expression
bdbi	⊢ BOUNDED (𝜑 ↔ 𝜓)

Proof of Theorem bdbi

Step	Hyp	Ref	Expression
1		bdbi.1	. . . 4 ⊢ BOUNDED 𝜑
2		bdbi.2	. . . 4 ⊢ BOUNDED 𝜓
3	1, 2	ax-bdim 17006	. . 3 ⊢ BOUNDED (𝜑 → 𝜓)
4	2, 1	ax-bdim 17006	. . 3 ⊢ BOUNDED (𝜓 → 𝜑)
5	3, 4	ax-bdan 17007	. 2 ⊢ BOUNDED ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))
6		dfbi2 392	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
7	5, 6	bd0r 17017	1 ⊢ BOUNDED (𝜑 ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ 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  ax-bdim 17006  ax-bdan 17007

This proof depends on definitions:  df-bi 117

This theorem is used by: (None)