Theorem bdbi 14663

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 14651	. . 3 ⊢ BOUNDED (𝜑 → 𝜓)
4	2, 1	ax-bdim 14651	. . 3 ⊢ BOUNDED (𝜓 → 𝜑)
5	3, 4	ax-bdan 14652	. 2 ⊢ BOUNDED ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))
6		dfbi2 388	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
7	5, 6	bd0r 14662	1 ⊢ BOUNDED (𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  BOUNDED wbd 14649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 14650  ax-bdim 14651  ax-bdan 14652

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)